Jitter decomposition method and measurement instrument

ABSTRACT

A jitter decomposition method for decomposing several jitter and noise components contained in an input signal, wherein the input signal is generated by a signal source, is disclosed. The jitter decomposition method comprises: receiving the input signal; at least one of determining and receiving a reconstructed data dependent jitter signal; at least one of determining and receiving an impulse response, the impulse response being associated with at least the signal source; and determining at least a first statistical parameter being associated with a first jitter component or a first noise component in the input signal and a second statistical parameter being associated with a second jitter component or a second noise component in the input signal, the second jitter component or the second noise component being different from the first jitter component or the first noise component, respectively. The first statistical parameter and the second statistical parameter are determined by applying a statistical method at two different times, wherein the first statistical parameter and the second statistical parameter are each determined based on at least one of the reconstructed data dependent jitter signal and the impulse response. Moreover, a measurement instrument is disclosed.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/795,931, filed Jan. 23, 2019, and U.S. Provisional Application No.62/799,478, filed Jan. 31, 2019, the disclosures of which areincorporated herein in their entirety.

FIELD OF THE DISCLOSURE

Embodiments of the present disclosure generally relate to a jitterdecomposition method for decomposing several jitter and noise componentscontained in an input signal. Further, embodiments of the presentdisclosure generally relate to a measurement instrument.

BACKGROUND

For jitter analysis, the components of jitter such as Data DependentJitter (DDJ), Periodic Jitter (PJ), Other Bounded Uncorrelated Jitter(OBUJ) and Random Jitter (RJ) must be separated and the bit error rate(BER) must be calculated.

So far, techniques are known that exclusively relate on determining theTime Interval Error (TIE) of the Total Jitter (TJ). In fact, the causesof the different jitter types lead to a distortion of the receivedsignal and they, therefore, have an influence on the TIE via thereceived signal. Accordingly, the respective components of jitter arecalculated once the Time Interval Error (TIE) of the Total Jitter (TJ)is determined.

However, it turned out that the measurement time is long if a highaccuracy is to be achieved. Put another way, the signal length of thesignal to be analyzed is long resulting in a long measuring duration ifhigh precision is aimed for.

Moreover, the respective components of jitter are obtained by averagingoperations. For instance, the Data Dependent Jitter (DDJ) is estimatedby averaging the Time Interval Error (TIE) of the Total Jitter (TJ),namely a DDJ eye diagram or the DDJ worst case eye diagram. Certaincomponents of jitter cannot be determined in a reliable manner.

Further, there is a cross talk between jitter and noise, as amplitudeperturbations may also cause jitter and temporal perturbations may alsocause noise. For an optimal accuracy, it is important to also analyzethis cross talk between jitter and noise.

Accordingly, there is a need for a fast and reliable possibility todecompose several jitter and noise components of an input signal.

SUMMARY

Embodiments of the present disclosure provide a jitter decompositionmethod for decomposing several jitter and noise components contained inan input signal, wherein the input signal is generated by a signalsource. In an embodiment, the method comprises the following steps:

receiving the input signal;

at least one of determining and receiving a reconstructed data dependentjitter signal;

at least one of determining and receiving an impulse response, theimpulse response being associated with at least the signal source; and

determining at least a first statistical parameter being associated witha first jitter component or a first noise component in the input signaland a second statistical parameter being associated with a second jittercomponent or a second noise component in the input signal, the secondjitter component or the second noise component being different from thefirst jitter component or the first noise component, respectively,

wherein the first statistical parameter and the second statisticalparameter are determined by applying a statistical method at least attwo different times, and wherein the first statistical parameter and thesecond statistical parameter are each determined based on at least oneof the reconstructed data dependent jitter signal and the impulseresponse.

Thus, the jitter decomposition method according to the presentdisclosure provides a possibility to jointly analyze two differentjitter components, two different noise components and/or a jittercomponent and a noise component. Thus, crosstalk between the individualjitter components and/or noise components can be analyzed and theindividual components can be determined more accurately.

The method according to the disclosure is based on the rationale toapply the statistical method at least at two different times. This way,a system of equations with at least two equations for the firststatistical parameter and the second statistical parameter is obtained.By solving this system of equations, the first statistical parameter andthe second statistical parameter can be determined.

The statistical method may be applied at more than two different times.This way, an overdetermined system of equations is obtained, The firstand the second statistical parameter may then be determined as thesolutions of the system of equations giving the minimum overall error,for example by applying a least mean squares approach.

As already explained above, there may be cross-talk between theperturbations in time, i.e. the jitter, and the perturbations inamplitude, i.e. the noise. Put another way, jitter may be caused by“horizontal” temporal perturbations, which will be denoted by “(h)” inthe following, and/or by “vertical” amplitude perturbations, which willbe denoted by a “(v)” in the following.

Likewise, noise may be caused by “horizontal” temporal perturbations,which will be denoted by “(h)” in the following, and/or by “vertical”amplitude perturbations, which will be denoted by a “(v)” in thefollowing.

In detail, the terminology used above and below is the following:

Horizontal periodic jitter PJ(h) is periodic jitter that is caused by atemporal perturbation.

Vertical periodic jitter PJ(v) is periodic jitter that is caused by anamplitude perturbation.

Horizontal other bounded uncorrelated jitter OBUJ(h) is other boundeduncorrelated jitter that is caused by a temporal perturbation.

Vertical other bounded uncorrelated jitter OBUJ(v) is other boundeduncorrelated jitter that is caused by an amplitude perturbation.

Horizontal random jitter RJ(h) is random jitter that is caused by atemporal perturbation.

Vertical random jitter RJ(v) is random jitter that is caused by anamplitude perturbation.

The definitions for noise are analogous to those for jitter:

Horizontal periodic noise PN(h) is periodic noise that is caused by atemporal perturbation.

Vertical periodic noise PN(v) is periodic noise that is caused by anamplitude perturbation.

Horizontal other bounded uncorrelated noise OBUN(h) is other boundeduncorrelated noise that is caused by a temporal perturbation.

Vertical other bounded uncorrelated noise OBUN(v) is other boundeduncorrelated noise that is caused by an amplitude perturbation.

Horizontal random noise RN(h) is random noise that is caused by atemporal perturbation.

Vertical random noise RN(v) is random noise that is caused by anamplitude perturbation.

The vertical random noise and the horizontal random noise components arestatistically independent from one another. Likewise, the horizontalrandom jitter and the vertical random jitter components arestatistically independent from one another. Moreover, all of thesecomponents are normal-distributed and have an expected value of zero.Thus, a respective variance of the components is the only quantityneeded to describe the distributions of the horizontal random jitter,the vertical random jitter, the horizontal random noise and the verticalrandom noise, respectively.

Accordingly, the first statistical parameter and/or the secondstatistical parameter may be a variance of the respective jitter ornoise component.

According to one aspect of the present disclosure, the first statisticalparameter is associated with a vertical jitter component and the secondstatistical parameter is associated with a horizontal jitter component,wherein the vertical jitter component is caused by an amplitudeperturbation, and wherein the horizontal jitter component is caused by atemporal perturbation. Thus, the vertical jitter and the horizontaljitter can be analyzed at the same time as the first statisticalparameter and the second statistical parameter are determinedsimultaneously by applying the statistical method at least at twodifferent times.

According to a further aspect of the present disclosure, the firststatistical parameter is associated with a noise component and thesecond statistical parameter is associated with a jitter component.Thus, the noise component and the jitter component are analyzed jointlyas the first statistical parameter and the second statistical parameterare determined simultaneously by applying the statistical method atleast at two different times. Accordingly, a joint jitter and noiseanalysis method is provided by the jitter decomposition method accordingto the disclosure.

In other words, the method according to the disclosure provides apossibility to decompose the several jitter and noise components, inparticular at least the vertical random noise, the horizontal randomnoise, the vertical random jitter and/or the horizontal random jitter,and to analyze them and their individual contributions e.g. to the timeinterval error.

In one embodiment of the present disclosure, at least one jittercomponent in the input signal is neglected or set to zero, or at leastone noise component is neglected or set to zero. As the at least onejitter component or the at least one noise component in the input signalis set to zero and/or neglected, the computational cost of the jitterdecomposition method is reduced and/or the method may be performed withless computational time.

In another embodiment of the present disclosure, a vertical periodicnoise signal in the input signal is neglected. As the vertical periodicnoise in the input signal is neglected, the computational cost of thejitter decomposition method is reduced and/or the method may beperformed with less computational time.

According to a further aspect, the first statistical parameter and thesecond statistical parameter comprise a statistical moment of secondorder or higher. For example, the first statistical parameter and/or thesecond statistical parameter comprise a variance. As already explainedabove, the respective variance of the components is the only quantityneeded to describe the distributions of the horizontal random jitter,the vertical random jitter, the horizontal random noise and the verticalrandom noise, respectively.

In another embodiment of the present disclosure, the input signal isdecoded, thereby generating a decoded input signal. In other words, theinput signal is divided into the individual symbol intervals and thevalues of the individual symbols (“bits”) are determined. Accordingly,the steps of the jitter decomposition method outlined above may also beperformed based on the decoded input signal instead of the input signal.

The step of decoding the input signal may be skipped if the input signalcomprises an already known bit sequence. For example, the input signalmay be a standardized signal such as a test signal that is determined bya communication protocol. In this case, the input signal does not needto be decoded, as the bit sequence contained in the input signal isalready known.

According to a further embodiment of the present disclosure, the secondstatistical parameter is determined based on at least the firststatistical parameter. Accordingly, the first statistical parameter maybe determined by applying the statistical method at first, and thesecond statistical parameter may afterwards be determined based on thealready determined first statistical parameter.

According to another aspect of the present disclosure, a thirdstatistical parameter associated with a third jitter component or athird noise component in the input signal is determined, the thirdjitter component or the third noise component being different from thefirst jitter component or the first noise component as well as thesecond jitter component or the second noise component. Thus, the jitterdecomposition method according to the present disclosure provides apossibility to jointly analyze three different jitter components, threedifferent noise components, two jitter components and a noise componentand/or one jitter component and three noise components. As explainedabove, crosstalk between the individual jitter components and/or thenoise components can be analyzed and the individual components can bedetermined more accurately.

A reconstructed vertical periodic noise signal is at least one ofdetermined and received, wherein the vertical periodic noise signal isassociated with periodic noise that is caused by an amplitudeperturbation. The first statistical parameter and/or the secondstatistical parameter may be determined at least partially based on thereconstructed vertical periodic noise signal.

According to one aspect of the present disclosure, a horizontal periodicjitter signal is determined based on at least one of the input signal,the reconstructed data dependent jitter signal and a reconstructedvertical periodic noise signal, wherein the horizontal periodic jittersignal is associated with periodic jitter that is caused by a temporalperturbation. The first statistical parameter and/or the secondstatistical parameter may be determined at least partially based on thedetermined horizontal periodic jitter signal.

In another embodiment of the present disclosure, a random perturbationsignal is determined based on at least one of the input signal, thereconstructed data dependent jitter signal, a reconstructed verticalperiodic noise signal and a determined horizontal periodic jittersignal. The first statistical parameter and/or the second statisticalparameter may be determined at least partially based on the randomperturbation signal.

The second statistical parameter may correspond to a statisticalparameter assigned to at least one of the horizontal random jitter andthe vertical random jitter, wherein the second statistical parameter isdetermined based on the random perturbation signal, wherein thestatistical parameter assigned to the horizontal random jitter isassociated with random jitter that is caused by a temporal perturbation,and wherein the statistical parameter assigned to the vertical randomjitter is associated with random jitter that is caused by an amplitudeperturbation.

The first statistical parameter may correspond to a statisticalparameter assigned to the vertical random noise, wherein the firststatistical parameter is determined based on the random perturbationsignal, the first statistical parameter being associated with a verticalrandom noise component of the input signal, wherein the verticalperiodic noise component is associated with periodic noise that iscaused by a temporal perturbation.

According to a further aspect of the present disclosure, the secondstatistical parameter corresponds to a statistical parameter assigned toat least one of the horizontal random jitter and the vertical randomjitter, wherein the second statistical parameter is determined based onat least one of the random perturbation signal and the determined firststatistical parameter, wherein the statistical parameter assigned to thehorizontal random jitter is associated with random jitter that is causedby a temporal perturbation, and wherein the statistical parameterassigned to the vertical random jitter is associated with random jitterthat is caused by an amplitude perturbation.

According to an embodiment of the disclosure, at least one of a timevariant equalizer filter and a time invariant equalizer filter isapplied in order to determine the horizontal periodic jitter signal. Thehorizontal periodic jitter component is distorted by a bit sequencecontained in the input signal, which distortion is removed by the timevariant equalizer filter and/or the time invariant equalizer filter. Insome embodiments, the time variant equalizer filter is used when theinput signal comprises duty cycle distortion and/or a duty cycledistortion comprised in the input signal is not negligible compared tothe horizontal periodic jitter. Accordingly, the time invariantequalizer filter may be used when the input signal is free of duty cycledistortion and/or a duty cycle distortion comprised in the input signalis negligible compared to the horizontal periodic jitter.

According to an aspect of the disclosure, horizontal periodic jittersignal parameters are determined in order to determine the horizontalperiodic jitter signal, the horizontal periodic jitter signal parametersbeing associated with periodic functions. The horizontal periodic jittersignal may be modelled as a sum over several sine-shaped functions,wherein the signal parameters associated with the periodic functions maybe the respective amplitudes, frequencies and/or phases of theindividual summands of the sum. In other words, the horizontal periodicjitter signal is modelled as a Fourier series, in particular as a finiteFourier series.

According to a further aspect, the horizontal periodic jitter signalparameters are determined jointly. In other words, the horizontalperiodic jitter signal parameters are determined simultaneously insteadof consecutively. In general, the accuracy of a joint determination ofseveral parameters is better than the consecutive determination of theseparameters. Thus, the accuracy of the determined horizontal periodicjitter signal parameters is enhanced by the joint determination.

The horizontal periodic jitter signal parameters may be determined by atleast one of minimizing and maximizing a cost functional. In otherwords, the cost functional depends on the horizontal periodic jittersignal parameters, which are determined such that the cost functionalbecomes minimal or maximal, respectively. Whether the cost functional isminimized or maximized depends on the particular definition of the costfunctional, as both cases can be converted into one another by a globalmultiplication of the cost functional with minus one. However, the costfunctional may be defined such that the deconvolution is performed byminimizing the cost functional, which can be regarded as the intuitivedefinition of the cost functional.

According to another aspect of the present disclosure, the reconstructeddata dependent jitter signal, the reconstructed vertical periodic noisesignal and the determined horizontal periodic jitter signal aresubtracted from the input signal in order to determine the randomperturbation signal. Accordingly, the random perturbation signal is freeof the above-mentioned components and thus approximately only containshorizontal random jitter and vertical random noise.

In another embodiment of the disclosure, a statistical analysis of therandom perturbation signal is performed in order to determine at leastone of the first statistical parameter and the second statisticalparameter. As explained above, the random perturbation signal containsapproximately only horizontal random jitter and vertical random noise,which both are normal-distributed quantities. Accordingly, a statisticalanalysis is suitable for the determination of the first statisticalparameter and/or the second statistical parameter.

According to another aspect, an expected value of the randomperturbation signal squared is determined in order to determine at leastone of the first statistical parameter and the second statisticalparameter. It turned out that the expected value squared is dependent onboth the random jitter variance and the vertical random noise variance,in particular linearly dependent. Thus, also the vertical random noisevariance may be determined from the determined expected value.

Moreover, the input signal may be PAM-n coded, wherein n is an integerbigger than 1. Accordingly, the method is not limited to binary signals(PAM-2 coded) since any kind of pulse-amplitude modulated signals may beprocessed.

Embodiments of the present disclosure further provide a jitterdecomposition method for decomposing several jitter and noise componentscontained in an input signal, wherein the input signal is generated by asignal source. In an embodiment, the method comprises the followingsteps:

receiving the input signal;

decoding the input signal, thereby generating a decoded input signal;

at least one of determining and receiving a reconstructed data dependentjitter signal;

at least one of determining and receiving a reconstructed verticalperiodic noise signal, wherein the vertical periodic noise signal isassociated with periodic noise that is caused by an amplitudeperturbation;

at least one of determining and receiving an impulse response, theimpulse response being associated with at least the signal source;

determining a horizontal periodic jitter signal based on at least one ofthe input signal, the decoded input signal, the reconstructed datadependent jitter signal and the reconstructed vertical periodic noisesignal, wherein the horizontal periodic jitter signal is associated withperiodic jitter that is caused by a temporal perturbation;

determining a random perturbation signal based on at least one of theinput signal, the decoded input signal, the reconstructed data dependentjitter signal, the reconstructed vertical periodic noise signal and thedetermined horizontal periodic jitter signal;

determining a vertical random noise variance based on the randomperturbation signal, the random noise variance being associated with avertical random noise component of the input signal, wherein thevertical periodic noise component is associated with periodic noise thatis caused by a temporal perturbation; and determining at least one of ahorizontal random jitter variance and a vertical random jitter variancebased on at least one of the random perturbation signal and thedetermined vertical random noise variance, wherein the horizontal randomjitter variance is associated with random jitter that is caused by atemporal perturbation, and wherein the vertical random jitter varianceis associated with random jitter that is caused by an amplitudeperturbation.

The explanations and advantages given above concerning the otherembodiments of the jitter decomposition method also apply to thisparticular embodiment.

Embodiments of the present disclosure further provide a measurementinstrument, comprising at least one input channel and an analysiscircuit or module being connected to the at least one input channel. Themeasurement instrument is configured to receive an input signal via theinput channel and to forward the input signal to the analysis module.The analysis module is configured to at least one of determine andreceive a reconstructed data dependent jitter signal. The analysismodule is configured to at least one of determine and receive an impulseresponse, the impulse response being associated with at least the signalsource. The analysis module is configured to determine at least a firststatistical parameter being associated with a first jitter component ora first noise component in the input signal and a second statisticalparameter being associated with a second jitter component or a secondnoise component in the input signal. The first statistical parameter andthe second statistical parameter are determined by applying astatistical method at two different times, wherein the first statisticalparameter and the second statistical parameter are each determined basedon at least one of the reconstructed data dependent jitter signal andthe impulse response.

Thus, the measurement instrument according to the present disclosureprovides a possibility to jointly analyze two different jittercomponents, two different noise components and/or a jitter component anda noise component. Thus, crosstalk between the individual jittercomponents and/or noise components can be analyzed by the measurementinstrument and the individual components can be determined moreaccurately.

The measurement instrument according to the disclosure is based on therationale to apply the statistical method at least at two differenttimes. This way, a system of equations with at least two equations forthe first statistical parameter and the second statistical parameter isobtained. By solving this system of equations, the first statisticalparameter and the second statistical parameter can be determined.

The measurement instrument may be configured to apply the statisticalmethod at more than two different times. This way, an overdeterminedsystem of equations is obtained, The first and the second statisticalparameter may then be determined as the solutions of the system ofequations giving the minimum overall error, for example by applying aleast mean squares approach.

In some embodiments, the measurement instrument is configured to performat least one of the jitter decomposition methods described above, forexample all of them.

According to one aspect of the present disclosure, the first statisticalparameter is associated with a vertical jitter component and the secondstatistical parameter is associated with a horizontal jitter component,wherein the vertical jitter component is caused by an amplitudeperturbation, and wherein the horizontal jitter component is caused by atemporal perturbation. Thus, the vertical jitter and the horizontaljitter can be analyzed at the same time as the first statisticalparameter and the second statistical parameter are determinedsimultaneously by applying the statistical method at least at twodifferent times.

According to a further embodiment of the present disclosure, the firststatistical parameter is associated with a noise component and thesecond statistical parameter is associated with a jitter component.Thus, the noise component and the jitter component can be analyzed atthe same time as the first statistical parameter and the secondstatistical parameter are determined simultaneously by applying thestatistical method at least at two different times.

The analysis module may be configured to neglect or set to zero at leastone of a jitter component in the input signal, noise component in theinput signal and a vertical periodic noise signal in the input signal.As the at least one jitter component or the at least one noise componentin the input signal is set to zero and/or neglected, the computationalcost of the jitter decomposition is reduced and/or the jitterdecomposition may be performed with less computational time.

In another embodiment of the present disclosure, the first statisticalparameter and the second statistical parameter comprise a statisticalmoment of second order or higher. For example, the first statisticalparameter and/or the second statistical parameter comprise a variance.As already explained above, the respective variance of the components isthe only quantity needed to describe the distributions of the horizontalrandom jitter, the vertical random jitter, the horizontal random noiseand the vertical random noise, respectively.

In some embodiments, the analysis module is configured to decode theinput signal, thereby generating a decoded input signal. In other words,the input signal is divided into the individual symbol intervals and thevalues of the individual symbols (“bits”) are determined. Accordingly,the measurement instrument may perform the steps of the jitterdecomposition method outlined above based on the decoded input signalinstead of the input signal.

The step of decoding the input signal may be skipped if the input signalcomprises an already known bit sequence. For example, the input signalmay be a standardized signal such as a test signal that is determined bya communication protocol. In this case, the input signal does not needto be decoded, as the bit sequence contained in the input signal isalready known.

The analysis module can comprise hardware and/or software portions,hardware and/or software modules, etc. In some embodiments, themeasurement instrument is configured to perform the jitter determinationmethods described above. Some of the method steps may be implemented inhardware and/or in software. In some embodiments, one or more of themethod steps may be implemented, for example, only in software. In someembodiments, all of the method steps are implemented, for example, onlyin software. In some embodiments, the method steps are implemented onlyin hardware.

DESCRIPTION OF THE DRAWINGS

The foregoing aspects and many of the attendant advantages of theclaimed subject matter will become more readily appreciated as the samebecome better understood by reference to the following detaileddescription, when taken in conjunction with the accompanying drawings,wherein:

FIG. 1 schematically shows a representative measurement system with anexample measurement instrument according to an embodiment of thedisclosure;

FIG. 2 shows a tree diagram of different types of jitter and differenttypes of noise;

FIG. 3 shows a representative flow chart of a jitter determinationmethod according to an embodiment of the disclosure;

FIG. 4 shows a representative flow chart of a signal parameterdetermination method according to an embodiment of the disclosure;

FIGS. 5A-5D show example histograms of different components of a timeinterval error;

FIG. 6 shows a representative flow chart of a method for separatingrandom jitter and horizontal periodic jitter according to an embodimentof the disclosure;

FIGS. 7A and 7B show diagrams of jitter components plotted over time;

FIG. 8 shows a schematic representation of a representative method fordetermining an autocorrelation function of random jitter according to anembodiment of the disclosure;

FIG. 9 shows an overview of different autocorrelation functions ofjitter components;

FIG. 10 shows an overview of different power spectrum densities ofjitter components;

FIG. 11 shows an overview of a bit error rate determined, a measured biterror rate and a reference bit error rate;

FIG. 12 shows an overview of a mathematical scale transformation of theresults of FIG. 11; and

FIG. 13 shows an overview of probability densities of the random jitter,the other bounded uncorrelated jitter as well as a superposition ofboth.

DETAILED DESCRIPTION

The detailed description set forth below in connection with the appendeddrawings, where like numerals reference like elements, is intended as adescription of various embodiments of the disclosed subject matter andis not intended to represent the only embodiments. Each embodimentdescribed in this disclosure is provided merely as an example orillustration and should not be construed as preferred or advantageousover other embodiments. The illustrative examples provided herein arenot intended to be exhaustive or to limit the claimed subject matter tothe precise forms disclosed.

FIG. 1 schematically shows a measurement system 10 comprising ameasurement instrument 12 and a device under test 14. The measurementinstrument 12 comprises a probe 16, an input channel 18, an analysiscircuit or module 20 and a display 22.

The probe 16 is connected to the input channel 18 which in turn isconnected to the analysis module 20. The display 22 is connected to theanalysis module 20 and/or to the input channel 18 directly. Typically, ahousing is provided that encompasses at least the analysis module 20.

Generally, the measurement instrument 12 may comprise an oscilloscope,as a spectrum analyzer, as a vector network analyzer or as any otherkind of measurement device being configured to measure certainproperties of the device under test 14.

The device under test 14 comprises a signal source 24 as well as atransmission channel 26 connected to the signal source 24.

In general, the signal source 24 is configured to generate an electricalsignal that propagates via the transmission channel 26. In particular,the device under test 14 comprises a signal sink to which the signalgenerated by the signal source 24 propagates via the transmissionchannel 26.

More specifically, the signal source 24 generates the electrical signalthat is then transmitted via the transmission channel 26 and probed bythe probe 16, in particular a tip of the probe 16. In fact, theelectrical signal generated by the signal source 24 is forwarded via thetransmission channel 26 to a location where the probe 16, in particularits tip, can contact the device under test 14 in order to measure theinput signal. Thus, the electrical signal may generally be sensedbetween the signal source 24 and the signal sink assigned to the signalsource 24, wherein the electrical signal may also be probed at thesignal source 24 or the signal sink directly. Put another way, themeasurement instrument 12, for example the analysis module 20, receivesan input signal via the probe 16 that senses the electrical signal.

The input signal probed is forwarded to the analysis module 20 via theinput channel 18. The input signal is then processed and/or analyzed bythe analysis module 20 in order to determine the properties of thedevice under test 14.

Therein and in the following, the term “input signal” is understood tobe a collective term for all stages of the signal generated by thesignal source 24 that exist before the signal reaches the analysismodule 20. In other words, the input signal may be altered by thetransmission channel 26 and/or by other components of the device undertest 14 and/or of the measurement instrument 12 that process the inputsignal before it reaches the analysis module 20. Accordingly, the inputsignal relates to the signal that is received and analyzed by theanalysis module 20.

The input signal usually contains perturbations in the form of totaljitter (TJ) that is a perturbation in time and total noise (TN) that isa perturbation in amplitude. The total jitter and the total noise inturn each comprise several components. Note that the abbreviationsintroduced in parentheses will be used in the following.

As is shown in FIG. 2, the total jitter (TJ) is composed of randomjitter (RJ) and deterministic jitter (DJ), wherein the random jitter(RJ) is unbounded and randomly distributed, and wherein thedeterministic jitter (DJ) is bounded. The deterministic jitter (DJ)itself comprises data dependent jitter (DDJ), periodic jitter (PJ) andother bounded uncorrelated jitter (OBUJ).

The data dependent jitter is directly correlated with the input signal,for example directly correlated with signal edges in the input signal.The periodic jitter is uncorrelated with the input signal and comprisesperturbations that are periodic, for example in time. The other boundeduncorrelated jitter comprises all deterministic perturbations that areneither correlated with the input signal nor periodic. The datadependent jitter comprises up to two components, namely inter-symbolinterference (ISI) and duty cycle distortion (DCD).

Analogously, the total noise (TN) comprises random noise (RN) anddeterministic noise (DN), wherein the deterministic noise contains datadependent noise (DDN), periodic noise (PN) and other boundeduncorrelated noise (OBUN).

Similarly to the jitter, the data dependent noise is directly correlatedwith the input signal, in particular directly correlated with signaledges in the input signal. The periodic noise is uncorrelated with theinput signal and comprises perturbations that are periodic, for examplein amplitude. The other bounded uncorrelated noise comprises alldeterministic perturbations that are neither correlated with the inputsignal nor periodic.

The data dependent noise comprises up to two components, namelyinter-symbol interference (ISI) and duty cycle distortion (DCD).

In general, there is cross-talk between the perturbations in time andthe perturbations in amplitude. Put another way, jitter may be caused by“horizontal” temporal perturbations, which is denoted by “(h)” in FIG. 2and in the following, and/or by “vertical” amplitude perturbations,which is denoted by a “(v)” in FIG. 2 and in the following.

Likewise, noise may be caused by “horizontal” temporal perturbations,which is denoted by “(h)” in FIG. 2 and in the following, and/or by“vertical” amplitude perturbations, which is denoted by a “(v)” in FIG.2 and in the following.

In detail, the terminology used below is the following:

Horizontal periodic jitter PJ(h) is periodic jitter that is caused by atemporal perturbation.

Vertical periodic jitter PJ(v) is periodic jitter that is caused by anamplitude perturbation.

Horizontal other bounded uncorrelated jitter OBUJ(h) is other boundeduncorrelated jitter that is caused by a temporal perturbation.

Vertical other bounded uncorrelated jitter OBUJ(v) is other boundeduncorrelated jitter that is caused by an amplitude perturbation.

Horizontal random jitter RJ(h) is random jitter that is caused by atemporal perturbation.

Vertical random jitter RJ(v) is random jitter that is caused by anamplitude perturbation.

The definitions for noise are analogous to those for jitter:

Horizontal periodic noise PN(h) is periodic noise that is caused by atemporal perturbation.

Vertical periodic noise PN(v) is periodic noise that is caused by anamplitude perturbation.

Horizontal other bounded uncorrelated noise OBUN(h) is other boundeduncorrelated noise that is caused by a temporal perturbation.

Vertical other bounded uncorrelated noise OBUN(v) is other boundeduncorrelated noise that is caused by an amplitude perturbation.

Horizontal random noise RN(h) is random noise that is caused by atemporal perturbation.

Vertical random noise RN(v) is random noise that is caused by anamplitude perturbation.

As mentioned above, noise and jitter each may be caused by “horizontal”temporal perturbations and/or by “vertical” amplitude perturbations.

The measurement instrument 12 or rather the analysis module 20 isconfigured to perform the steps schematically shown in FIGS. 3, 4 and 6in order to analyze the jitter and/or noise components contained withinthe input signal, namely the jitter and/or noise components mentionedabove.

In some embodiments, one or more computer-readable storage media isprovided containing computer readable instructions embodied thereonthat, when executed by one or more computing devices (contained in orassociated with the measurement instrument 12, the analysis module 20,etc.), cause the one or more computing devices to perform one or moresteps of the method of FIGS. 3, 4, 6, and/or 8 described below. In otherembodiments, one or more of these method steps can be implemented indigital and/or analog circuitry or the like.

Model of the Input Signal

First of all, a mathematical substitute model of the input signal orrather of the jitter components and the noise components of the inputsignal is established. Without loss of generality, the input signal isassumed to be PAM-n coded in the following, wherein n is an integerbigger than 1. Hence, the input signal may be a binary signal (PAM-2coded).

Based on the categorization explained above with reference to FIG. 2,the input signal at a time t/T_(b) is modelled as

$\begin{matrix}{{x_{TN}\left( {t\text{/}T_{b}} \right)} = {{\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{{b(k)} \cdot {h\left( {{t\text{/}T_{b}} - k - {{ɛ(k)}\text{/}T_{b}}} \right)}}} + {\sum\limits_{i = 0}^{N_{{PN}{(v)}} - 1}\;{A_{i} \cdot {\sin\left( {{2{\pi \cdot f_{i}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i}} \right)}}} + {x_{{RN}{(v)}}\left( {t\text{/}T_{b}} \right)} + {{x_{{OBUN}{(v)}}\left( {t\text{/}T_{b}} \right)}.}}} & \left( {E{.1}} \right)\end{matrix}$

In the first term, namely

${\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{{b(k)} \cdot {h\left( {{t\text{/}T_{b}} - k - {{ɛ(k)}\text{/}T_{b}}} \right)}}},$

b(k) represents a bit sequence sent by the signal source 24 via thetransmission channel 26, wherein T_(b) is the bit period.

Note that strictly speaking the term “bit” is only correct for a PAM-2coded input signal. However, the term “bit” is to be understood to alsoinclude a corresponding signal symbol of the PAM-n coded input signalfor arbitrary integer n.

h(t/T_(b)) is the joint impulse response of the signal source 24 and thetransmission channel 26. In case of directly probing the signal source24, h(t/T_(b)) is the impulse response of the signal source 24 since notransmission channel 26 is provided or rather necessary.

Note that the joint impulse response h(t/T_(b)) does not comprisecontributions that are caused by the probe 16, as these contributionsare usually compensated by the measurement instrument 12 or the probe 16itself in a process called “de-embedding”. Moreover, contributions fromthe probe 16 to the joint impulse response h(t/T_(b)) may be negligiblecompared to contributions from the signal source 24 and the transmissionchannel 26.

N_(pre) and N_(post) respectively represent the number of bits beforeand after the current bit that perturb the input signal due tointer-symbol interference. As already mentioned, the lengthN_(pre)+N_(post)+1 may comprise several bits, for example severalhundred bits, especially in case of occurring reflections in thetransmission channel 26.

Further, ε(k) is a function describing the time perturbation, i.e. ε(k)represents the temporal jitter.

Moreover, the input signal also contains periodic noise perturbations,which are represented by the second term in equation (E.1), namely

${\sum\limits_{i = 0}^{N_{{PN}{(v)}} - 1}\;{A_{i} \cdot {\sin\left( {{2{\pi \cdot f_{i}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i}} \right)}}},$

The periodic noise perturbation is modelled by a series over N_(PN(v))sine-functions with respective amplitudes A_(i), frequencies f_(i) andphases ϕ_(i), which is equivalent to a Fourier series of the verticalperiodic noise.

The last two terms in equation (E.1), namely+x _(RN(v))(t/T _(b))+x _(OBUN(v))(t/T _(b)),

represent the vertical random noise and the vertical other boundeduncorrelated noise contained in the input signal, respectively.

The function ε(k) describing the temporal jitter is modelled as follows:

$\begin{matrix}{{{ɛ(k)}\text{/}T_{b}} = {{\sum\limits_{i = 0}^{N_{{PJ}{(h)}} - 1}\;{a_{i}\text{/}{T_{b} \cdot {\sin\left( {{2{\pi \cdot \vartheta_{i}}\text{/}{f_{b} \cdot k}} + \varphi_{i}} \right)}}}} + {{ɛ_{RJ}(k)}\text{/}T_{b}} + {{ɛ_{OBUJ}(k)}\text{/}{T_{b}.}}}} & \left( {E{.2}} \right)\end{matrix}$

The first term in equation (E.2), namely

${\sum\limits_{i = 0}^{N_{{PJ}{(h)}} - 1}\;{a_{i}\text{/}{T_{b} \cdot {\sin\left( {{2{\pi \cdot \vartheta_{i}}\text{/}{f_{b} \cdot k}} + \varphi_{i}} \right)}}}},$

represents the periodic jitter components that are modelled by a seriesover N_(PJ(h)) sine-functions with respective amplitudes a_(i),frequencies ϑ_(i) and phases φ_(i), which is equivalent to a Fourierseries of the horizontal periodic jitter.

The last two terms in equation (E.2), namelyε_(RJ)(k)/T _(b)+ε_(OBUJ)(k)/T _(b),

represent the random jitter and the other bounded uncorrelated jittercontained in the total jitter, respectively.

In order to model duty cycle distortion (DCD), the model of (E.1) has tobe adapted to depend on the joint step response h_(s)(t/T_(b),b(k)) ofthe signal source 24 and the transmission channel 26.

As mentioned earlier, the step response h_(s)(t/T_(b),b(k)) of thesignal source 24 may be taken into account provided that the inputsignal is probed at the signal source 24 directly.

Generally, duty cycle distortion (DCD) occurs when the step response fora rising edge signal is different to the one for a falling edge signal.

The inter-symbol interference relates, for example, to limitedtransmission channel or rather reflection in the transmission.

The adapted model of the input signal due to the respective stepresponse is given by

$\begin{matrix}{{x_{TN}\left( {t\text{/}T_{b}} \right)} = {{\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h_{s}\left( {{{t\text{/}T_{b}} - k - {{ɛ(k)}\text{/}T_{b}}},{b(k)}} \right)}}} + x_{- \infty} + {\sum\limits_{i = 0}^{{N_{PN}{(v)}} - 1}\;{A_{i} \cdot {\sin\left( {{2{\pi \cdot f_{i}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i}} \right)}}} + {x_{{RN}{(v)}}\left( {t\text{/}T_{b}} \right)} + {{x_{{OBUN}{(v)}}\left( {t\text{/}T_{b}} \right)}.}}} & \left( {E{.3}} \right)\end{matrix}$

Therein, x_(−∞) represents the state at the start of the transmission ofthe input signal, for example the state of the signal source 24 and thetransmission channel 26 at the start of the transmission of the inputsignal.

The step response h_(s)(t/T_(b),b(k)) depends on the bit sequence b(k),or more precisely on a sequence of N_(DCD) bits of the bit sequenceb(k), wherein N_(DCD) is an integer bigger than 1.

Note that there is an alternative formulation of the duty cycledistortion that employs N_(DCD)=1. This formulation, however, is a meremathematical reformulation of the same problem and thus equivalent tothe present disclosure.

Accordingly, the step response h_(s)(t/T_(b),b(k)) may generally dependon a sequence of N_(DCD) bits of the bit sequence b(k), wherein N_(DCD)is an integer value.

Typically, the dependency of the step response h_(s)(t/T_(b),b(k)) onthe bit sequence b (k) ranges only over a few bits, for instanceN_(DCD)=2, 3, . . . , 6.

For N_(DCD)=2 this is known as “double edge response (DER)”, while forN_(DCD)>2 this is known as “multi edge response (MER)”.

Without restriction of generality, the case N_(DCD)=2 is described inthe following. However, the outlined steps also apply to the caseN_(DCD)>2 with the appropriate changes. As indicated above, thefollowing may also be (mathematically) reformulated for N_(DCD)=1.

In equation (E.3), the term b(k)−b(k−1), which is multiplied with thestep response h_(s)(t/T_(b),b(k)), takes two subsequent bit sequences,namely b(k) and b(k−1), into account such that a certain signal edge isencompassed.

In general, there may be two different values for the step responseh_(s)(t/T_(b),b(k)), namely h_(s) ^((r))(t/T_(b)) for a rising signaledge and h_(s) ^((f))(t/T_(b)) for a falling signal edge. In otherwords, the step response h_(s)(t/T_(b),b(k)) may take the following twovalues:

$\begin{matrix}{{h_{s}\left( {{t\text{/}T_{b}},{b(k)}} \right)} = \left\{ \begin{matrix}{{h_{s}^{(r)}\left( {t\text{/}T_{b}} \right)},} & {{{b(k)} - {b\left( {k - 1} \right)}} \geq 0} \\{{h_{s}^{(f)}\left( {t\text{/}T_{b}} \right)},} & {{{b(k)} - {b\left( {k - 1} \right)}} < 0.}\end{matrix} \right.} & \left( {E{.4}} \right)\end{matrix}$

If the temporal jitter ε(k) is small, equation (E.3) can be linearizedand then becomes

$\begin{matrix}{{x_{TN}\left( {t\text{/}T_{b}} \right)} \approx {{\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot h_{s}}\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}} + x_{- \infty} + {\sum\limits_{i = 0}^{N_{{PN}{(v)}} - 1}\;{A_{i} \cdot {\sin\left( {{2{\pi \cdot f_{i}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i}} \right)}}} + {x_{{RN}{(v)}}\left( {t\text{/}T_{b}} \right)} + {x_{{OBUN}{(v)}}\left( {t\text{/}T_{b}} \right)} - {\sum\limits_{k = {- N_{pre}}}^{N_{post}}{{ɛ(k)}\text{/}{T_{b} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {{h\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}.}}}}}} & \left( {E{.5}} \right)\end{matrix}$

Note that the last term in equation (E.5), namely

${\sum\limits_{k = {- N_{pre}}}^{N_{post}}{{ɛ(k)}\text{/}{T_{b} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}}}},$

describes an amplitude perturbation that is caused by the temporaljitter ε(k).

It is to be noted that the input signal comprises the total jitter aswell as the total noise so that the input signal may also be labelled byx_(TJ)(t/T_(b)).

Clock Data Recovery

A clock data recovery is now performed based on the received inputsignal employing a clock timing model of the input signal, which clocktiming model is a slightly modified version of the substitute modelexplained above. The clock timing model will be explained in more detailbelow.

Generally, the clock signal T_(clk) is determined while simultaneouslydetermining the bit period T_(b) from the times t_(edge)(i) of signaledges based on the received input signal.

More precisely, the bit period T_(b) scaled by the sampling rate 1/T_(a)is inter alia determined by the analysis module 20.

In the following, {circumflex over (T)}_(b)/T_(a) is understood to bethe bit period that is determined by the analysis module 20. The symbol“{circumflex over ( )}” marks quantities that are determined by theanalysis module 20, in particular quantities that are estimated by theanalysis module 20.

One aim of the clock data recovery is to also determine a time intervalerror TIE(k) caused by the different types of perturbations explainedabove.

Moreover, the clock data recovery may also be used for decoding theinput signal, for determining the step response h(t\T_(b)) and/or forreconstructing the input signal. Each of these applications will beexplained in more detail below.

Note that for each of these applications, the same clock data recoverymay be performed. Alternatively, a different type of clock data recoverymay be performed for at least one of these applications.

In order to enhance the precision or rather accuracy of the clock datarecovery, the bit period {circumflex over (T)}_(b)/T_(a) is determinedjointly with at least one of the deterministic jitter componentsmentioned above and with a deviation Δ{circumflex over (T)}_(b)/T_(a)from the bit period {circumflex over (T)}_(b)/T_(a).

In the case described in the following, the bit period {circumflex over(T)}_(b)/T_(a) and the deviation Δ{circumflex over (T)}_(b)/T_(a) areestimated together with the data dependent jitter component and theperiodic jitter components. Therefore, the respective jitter componentsare taken into account when providing a cost functional that is to beminimized.

The principle of minimizing a cost functional, also called criterion, inorder to determine the clock signal T_(clk) is known.

More precisely, the bit period {circumflex over (T)}_(b)/T_(a) and thedeviation Δ{circumflex over (T)}_(b)/T_(a) are determined by determiningthe times t_(edge)(i) of signal edges based on the received input signaland by then minimizing the following cost functional K, for example byemploying a least mean squares approach:

$\begin{matrix}{\left. {K = {\sum\limits_{i = 0}^{N - 1}\;{\left\lbrack {\frac{t_{edge}(i)}{T_{a}} - {k_{i,\eta} \cdot \frac{{\hat{T}}_{b}(\eta)}{T_{a}}} - \frac{\Delta\;{{\hat{T}}_{b}(\eta)}}{T_{a}} - {\sum\limits_{L_{{ISI}_{pre}}}^{L_{{ISI}_{post}}}{{\hat{h}}_{r,f}\left( {{k_{i} - \xi},{b\left( k_{i} \right)},{b\left( {k_{i} - 1} \right)},{b\left( {k_{i} - \xi} \right)},{b\left( {k_{i} - \xi - 1} \right)}} \right)}} -}\quad \right.{\quad\quad}{\sum\limits_{\mu = 0}^{M_{PJ} - 1}\;{{{\hat{C}}_{\mu} \cdot {\quad\quad}}{\sin\left( {{2{\pi \cdot {\hat{v}}_{\mu}}\text{/}{T_{a} \cdot k_{i}}} + \Psi_{\mu}} \right)}}}}}} \right\rbrack^{2}.} & \left( {E{.6}} \right)\end{matrix}$

As mentioned above, the cost functional K used by the method accordingto the present disclosure comprises terms concerning the data dependentjitter component, which is represented by the fourth term in equation(E.6) and the periodic jitter components, which are represented by thefifth term in equation (E.6), namely the vertical periodic jittercomponents and/or the horizontal periodic jitter components.

Therein, L_(ISI), namely the length L_(ISI) _(pre) +L_(ISI) _(post) , isthe length of an Inter-symbol Interference filter (ISI-filter)ĥ_(r,f)(k) that is known from the state of the art and that is used tomodel the data dependent jitter. The length L_(ISI) should be chosen tobe equal or longer than the length of the impulse response, namely theone of the signal source 24 and the transmission channel 26.

Hence, the cost functional K takes several signal perturbations intoaccount rather than assigning their influences to (random) distortionsas typically done in the prior art.

In fact, the term

$\sum\limits_{L_{{ISI}_{pre}}}^{L_{{ISI}_{post}}}{{\hat{h}}_{r,f}\left( {{k_{i} - \xi},{b\left( k_{i} \right)},{b\left( {k_{i} - 1} \right)},{b\left( {k_{i} - \xi} \right)},{b\left( {k_{i} - \xi - 1} \right)}} \right)}$

relates to the data dependent jitter component. The term assigned to thedata dependent jitter component has several arguments for improving theaccuracy since neighbored edge signals, also called aggressors, aretaken into account that influence the edge signal under investigation,also called victim.

In addition, the term

$\sum\limits_{\mu = 0}^{M_{PJ} - 1}\;{{\hat{C}}_{\mu} \cdot {\sin\left( {{2{\pi \cdot {\hat{v}}_{\mu}}\text{/}{T_{a} \cdot k_{i}}} + {\hat{\Psi}}_{\mu}} \right)}}$

concerns the periodic jitter components, namely the vertical periodicjitter components and/or the horizontal periodic jitter components, thatare also explicitly mentioned as described above. Put it another way, itis assumed that periodic perturbations occur in the received inputsignal which are taken into consideration appropriately.

If the signal source 24 is configured to perform spread spectrumclocking, then the bit period T_(b)/T_(a) is not constant but variesover time.

The bit period can then, as shown above, be written as a constantcentral bit period T_(b), namely a central bit period being constant intime, plus a deviation ΔT_(b) from the central bit period T_(b), whereinthe deviation ΔT_(b) varies over time.

In this case, the period of observation is divided into several timeslices or rather time sub-ranges. For ensuring the above concept, theseveral time slices are short such that the central bit period T_(b) isconstant in time.

The central bit period T_(b) and the deviation ΔT_(b) are determined forevery time slice or rather time sub-range by minimizing the followingcost functional K:

$\begin{matrix}{{K = {\sum\limits_{i = 0}^{N - 1}\;\left\lbrack {\frac{t_{edge}(i)}{T_{a}} - {k_{i,\eta} \cdot \frac{{\hat{T}}_{b}(\eta)}{T_{a}}} - \frac{\Delta\;{{\hat{T}}_{b}(\eta)}}{T_{a}} - {\sum\limits_{L_{{ISI}_{pre}}}^{L_{{ISI}_{post}}}{{\hat{h}}_{r,f}\left( {{k_{i} - \xi},{b\left( k_{i} \right)},{b\left( {k_{i} - 1} \right)},{b\left( {k_{i} - \xi} \right)},{b\left( {k_{i} - \xi - 1} \right)}} \right)}} - {\sum\limits_{\mu = 0}^{M_{PJ} - 1}\;{{\hat{C}}_{\mu} \cdot {\sin\left( {{2{\pi \cdot {\hat{v}}_{\mu}}\text{/}{T_{a} \cdot k_{i}}} + {\hat{\Psi}}_{\mu}} \right)}}}} \right\rbrack^{2}}},} & \left( {E{.7}} \right)\end{matrix}$

which is the same cost functional as the one in equation (E.6).

Based on the determined bit period {circumflex over (T)}_(b)/T_(a) andbased on the determined deviation Δ{circumflex over (T)}_(b)/T_(a), thetime interval error TIE(i)/T_(a) is determined asTIE(i)/T _(a) =t _(edge)(i)/T _(a) −k _(i,η) ·{circumflex over (T)}_(b)−(η)/T _(a) −Δ{circumflex over (T)} _(b)(η)/T _(a).

Put another way, the time interval error TIE(i)/T_(a) corresponds to thefirst three terms in equations (E.6) and (E.7), respectively.

However, one or more of the jitter components may also be incorporatedinto the definition of the time interval error TIE(i)/T_(a).

In the equation above regarding the time interval error TIE(i)/T_(a),the term k_(i,η)·{circumflex over (T)}_(b)(η)/T_(a)+Δ{circumflex over(T)}_(b)(η)/T_(a) represents the clock signal for the i-th signal edge.This relation can be rewritten as follows {circumflex over(T)}_(clk)=k_(i,η)·{circumflex over (T)}_(b)(η)/T_(a)+Δ{circumflex over(T)}_(b)(η)/T_(a).

As already described, a least mean squares approach is applied withwhich at least the constant central bit period T_(b) and the deviationΔT_(b) from the central bit period T_(b) are determined.

In other words, the time interval error TIE(i)/T_(a) is determined andthe clock signal T_(clk) is recovered by the analysis described above.

In particular, the total time interval error TIE_(TJ)(k) is determinedemploying the clock data recovery method described above (step S.3.1 inFIG. 3).

Generally, the precision or rather accuracy is improved since theoccurring perturbations are considered when determining the bit periodby determining the times t_(edge)(i) of signal edges based on thereceived input signal and by then minimizing the cost functional K.

Decoding the Input Signal

With the recovered clock signal T_(clk) determined by the clock recoveryanalysis described above, the input signal is divided into theindividual symbol intervals and the values of the individual symbols(“bits”) b(k) are determined.

The signal edges are assigned to respective symbol intervals due totheir times, namely the times t_(edge)(i) of signal edges. Usually, onlyone signal edge appears per symbol interval.

In other words, the input signal is decoded by the analysis module 20,thereby generating a decoded input signal. Thus, b(k) represents thedecoded input signal.

The step of decoding the input signal may be skipped if the input signalcomprises an already known bit sequence. For example, the input signalmay be a standardized signal such as a test signal that is determined bya communication protocol. In this case, the input signal does not needto be decoded, as the bit sequence contained in the input signal isalready known.

Joint Analysis of the Step Response and of the Periodic SignalComponents

The analysis module 20 is configured to jointly determine the stepresponse of the signal source 24 and the transmission channel 26 on onehand and the vertical periodic noise parameters defined in equation(E.5) on the other hand, wherein the vertical periodic noise parametersare the amplitudes A_(i), the frequencies f_(i) and the phases ϕ_(i)(step S.3.2 in FIG. 3).

Therein and in the following, the term “determine” is understood to meanthat the corresponding quantity may be computed and/or estimated with apredefined accuracy.

Thus, the term “jointly determined” also encompasses the meaning thatthe respective quantities are jointly estimated with a predefinedaccuracy.

However, the vertical periodic jitter parameters may also be jointlydetermined with the step response of the signal source 24 and thetransmission channel 26 in a similar manner.

The concept is generally called joint analysis method.

In general, the precision or rather accuracy is improved due to jointlydetermining the step response and the periodic signal components.

Put differently, the first three terms in equation (E.5), namely

${{\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h_{s}\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}}} + x_{- \infty} + {\sum\limits_{i = 0}^{N_{{PN}{(v)}} - 1}\;{A_{i} \cdot {\sin\left( {{2{\pi \cdot f_{i}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i}} \right)}}}},$

are jointly determined by the analysis module 20.

As a first step, the amplitudes A_(i), the frequencies f_(i) and thephases ϕ_(i) are roughly estimated via the steps depicted in FIG. 4.

First, a clock data recovery is performed based on the received inputsignal (step S.4.1), for example as described above.

Second, the input signal is decoded (step S.4.2).

Then, the step response, for example the one of the signal source 24 andthe transmission channel 26, is roughly estimated based on the decodedinput signal (step S.4.3), for example by matching the first term inequation (E.5) to the measured input signal, in particular via a leastmean squares approach.

Therein and in the following, the term “roughly estimated” is to beunderstood to mean that the corresponding quantity is estimated with anaccuracy being lower compared to the case if the quantity is determined.

Now, a data dependent jitter signal x_(DDJ) being a component of theinput signal only comprising data dependent jitter is reconstructedbased on the roughly estimated step response (step S.4.4).

The data dependent jitter signal x_(DDJ) is subtracted from the inputsignal (step S.4.5). The result of the subtraction is the signalx_(PN+RN) that approximately only contains periodic noise and randomnoise.

Finally, the periodic noise parameters A_(i), f_(i), ϕ_(i) are roughlyestimated based on the signal x_(PN+RN) (step S.4.6), in particular viaa fast Fourier transform of the signal x_(PN+RN).

In the following, these roughly estimated parameters are marked bysubscripts “0”, i.e. the rough estimates of the frequencies are f_(i,0)and the rough estimates of the phases are ϕ_(i,0). The roughly estimatedfrequencies f_(i,0) and phases ϕ_(i,0) correspond to working points forlinearizing purposes as shown hereinafter.

Accordingly, the frequencies and phases can be rewritten as follows:f _(i) /f _(b) =f _(i,0) /f _(b) +Δf _(i) /f _(b)ϕ_(i)=ϕ_(i,0)+Δϕ_(i)   (E.8)

Therein, Δf_(i) and Δϕ_(i) are deviations of the roughly estimatedfrequencies f_(i,0) and phases ϕ_(i,0) from the actual frequencies andphases, respectively. By construction, the deviations Δf_(i) and Δϕ_(i)are much smaller than the associated frequencies f_(i) and phases ϕ_(i),respectively.

With the re-parameterization above, the sine-function in the third termin equation (E.5), namely

$\sum\limits_{i = 0}^{N_{{PN}{(v)}} - 1}\;{A_{i} \cdot {\sin\left( {{2{\pi \cdot f_{i}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i}} \right)}}$

can be linearized as follows while using small-angle approximation orrather the Taylor series:

$\begin{matrix}\begin{matrix}{{A_{i} \cdot {\sin\left( {{2{\pi \cdot f_{i}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i}} \right)}} = {A_{i} \cdot {\sin\left( {{2\pi\; f_{i,0}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i,0} +} \right.}}} \\\left. {{2{\pi \cdot \Delta}\; f_{i}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + {\Delta\phi}_{i}} \right) \\{= {A_{i} \cdot \left\lbrack {{\sin\left( {{2{\pi \cdot f_{i,0}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i,0}} \right)} \cdot} \right.}} \\{{\cos\left( {{2{\pi \cdot \Delta}\; f_{i}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + {\Delta\phi}_{i}} \right)} +} \\{{{\cos\left( {{2{\pi \cdot f_{i,0}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i,0}} \right)} \cdot}\;} \\{\left. {\sin\left( {{2{\pi \cdot \Delta}\; f_{i}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + {\Delta\phi}_{i}} \right)} \right\rbrack\;} \\{\approx {{A_{i} \cdot {\sin\left( {{2{\pi \cdot f_{i,0}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i,0}} \right)}} +}} \\{{A_{i} \cdot {\cos\left( {{2\pi\; f_{i,0}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i,0}} \right)} \cdot}\;} \\{\left\lbrack {{2{\pi \cdot \Delta}\; f_{i}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + {\Delta\phi}_{i}} \right\rbrack\;} \\{= {{p_{i,0} \cdot {\sin\left( {{2{\pi \cdot f_{i,0}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i,0}} \right)}} +}} \\{{{p_{i,1} \cdot 2}{\pi \cdot t}\text{/}{T_{b} \cdot}}\;} \\{{{\cos\left( {{2{\pi \cdot f_{i,0}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i,0}} \right)} +}\;} \\{{p_{i,2} \cdot {{\cos\left( {{2{\pi \cdot f_{i,0}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + \phi_{i,0}} \right)}.}}\;}\end{matrix} & \left( {E{.9}} \right)\end{matrix}$

In the last two lines of equation (E.9), the following new, linearlyindependent parameters have been introduced, which are determinedafterwards:p _(i,0) =A _(i)p _(i,1) =A _(i) ·Δf _(i) /f _(b)p _(i,2) =A _(i)·Δϕ_(i)   (E.10)

With the mathematical substitute model of equation (E.5) adapted thatway, the analysis module 20 can now determine the step responseh_(s)(t/T_(b),b(k)), more precisely the step response h_(s)^((r))(t/T_(b)) for rising signal edges and the step response h_(s)^((f))(t/T_(b)) for falling signal edges, and the vertical periodicnoise parameters, namely the amplitudes A_(i), the frequencies f_(i) andthe phases ϕ_(i), jointly, i.e. at the same time.

This may be achieved by minimizing a corresponding cost functional K, inparticular by applying a least mean squares method to the costfunctional K. The cost functional has the following general form:K=[ A (k)· {circumflex over (x)} − x _(L) k)]^(T)·[ A (k)· {circumflexover (x)} − x _(L)(k)].   (E.11)

Therein, x _(L)(k) is a vector containing L measurement points of themeasured input signal. {circumflex over (x)} is a corresponding vectorof the input signal that is modelled as in the first three terms ofequation (E.5) and that is to be determined. A(k) is a matrix dependingon the parameters that are to be determined.

In some embodiments, matrix A(k) comprises weighting factors for theparameters to be determined that are assigned to the vector x _(L)(k).

Accordingly, the vector x _(L)(k) may be assigned to the step responseh_(s) ^((r))(t/T_(b)) for rising signal edges, the step response h_(s)^((f))(t/T_(b)) for falling signal edges as well as the verticalperiodic noise parameters, namely the amplitudes A_(i), the frequenciesf_(i) and the phases ϕ_(i).

The least squares approach explained above can be extended to aso-called maximum-likelihood approach. In this case, themaximum-likelihood estimator {circumflex over (x)} _(ML) is given by{circumflex over (x)} _(ML)=[ A ^(T)(k)· R _(n) ⁻¹(k)· A (k)]⁻¹·[ A^(T)(k)· R _(n) ⁻¹(k)· x _(L)(k)].   (E.11a)

Therein, R _(n)(k) is the covariance matrix of the perturbations, i.e.the jitter and noise components comprised in equation (E.5).

Note that for the case of pure additive white Gaussian noise, themaximum-likelihood approach is equivalent to the least squares approach.

The maximum-likelihood approach may be simplified by assuming that theperturbations are not correlated with each other. In this case, themaximum-likelihood estimator becomes{circumflex over (x)} _(ML)≈[ A ^(T)(k)·(( r _(n,i)(k)·1 ^(T))º A(k))]⁻¹·[ A ^(T)(k)·( r _(n,i)(k)º x _(L)(k))].   (E.11b)

Therein, 1 ^(T) is a unit vector and the vector r _(n,i)(k) comprisesthe inverse variances of the perturbations.

For the case that only vertical random noise and horizontal random noiseare considered as perturbations, this becomes

$\begin{matrix}{\left\lbrack {{\underset{\_}{r}}_{n,i}(k)} \right\rbrack_{l} = \left( {{\frac{\sigma_{\epsilon,{RJ}}^{2}}{T_{b}^{2}}{\sum\limits_{m = {- N_{post}}}^{N_{pre}}\;{\left\lbrack {{b\left( {k - l - m} \right)} - {b\left( {k - l - m - 1} \right)}} \right\rbrack^{2} \cdot \left( {h\left( {m,{b(m)}} \right)} \right)^{2}}}} + \sigma_{{RN}{(v)}}^{2}} \right)^{- 1}} & \left( {E{.11}c} \right)\end{matrix}$

Employing equation (E.11c) in equation (E.11b), an approximate maximumlikelihood estimator is obtained for the case of vertical random noiseand horizontal random noise being approximately Gaussian distributed.

If the input signal is established as a clock signal, i.e. if the valueof the individual symbol periodically alternates between two values withone certain period, the approaches described above need to be adapted.The reason for this is that the steps responses usually extend overseveral bits and therefore cannot be fully observed in the case of aclock signal. In this case, the quantities above have to be adapted inthe following way:{circumflex over (x)} [( ĥ _(s) ^((r)))^(T)( ĥ _(s) ^((f))))^(T){circumflex over (p)} _(3N) _(Pj) ^(T)]^(T)A (k)=[ b _(L,N) ^((r))(k)− b _(L,N) ^((r))(k−T _(b) /T _(a)) b _(L,N)^(f)(k)− b _(L,N) ^((f))(k−T _(b) /T _(a)) t _(L,3N) _(PJ) (k)]x _(L)(k)=[ b _(L,N) ^((r))(k)− b _(L,N) ^((r))(k−T _(b) /T _(a))]· h_(s) ^((r))+[ b _(L,N) ^((f))(k)− b _(L,N) ^((f))(k−T _(b) /T _(a))]· h_(s) ^((f)) +t _(L,3N) _(PJ) (k)· p _(3N) _(Pj) +n _(L)(k).   (E.11d)

Input Signal Reconstruction and Determination of Time Interval Error

With the determined step response and with the determined periodic noisesignal parameters, a reconstructed signal {circumflex over(x)}_(DDJ+PN(v))(t/T_(b)) containing only data dependent jitter andvertical periodic noise can be determined while taking equation (E.5)into account.

Thus, the reconstructed signal {circumflex over(x)}_(DDJ+PN(v))(t/T_(b)) is given by

$\begin{matrix}\begin{matrix}{{{\hat{x}}_{{DDJ} + {{PN}{(v)}}}\left( {t\text{/}T_{b}} \right)} = {{{\hat{x}}_{DDJ}\left( {t\text{/}T_{b}} \right)} + {{\hat{x}}_{{PN}{(v)}}\left( {t\text{/}T_{b}} \right)}}} \\{= {\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{\left\lbrack {{\hat{b}(k)} - {\hat{b}\left( {k - 1} \right)}} \right\rbrack \cdot}}} \\{{{\hat{h}}_{s}\left( {{{t\text{/}T_{b}} - k},{\hat{b}(k)}} \right)} + {\hat{x}}_{- \infty} +} \\{\sum\limits_{i = 0}^{N_{{PN}{(v)}} - 1}\;{{\hat{A}}_{i} \cdot {{\sin\left( {{2{\pi \cdot {\hat{f}}_{i}}\text{/}{f_{b} \cdot t}\text{/}T_{b}} + {\hat{\phi}}_{i}} \right)}.}}}\end{matrix} & \left( {E{.12}} \right)\end{matrix}$

Moreover, also a reconstructed signal {circumflex over(x)}_(DDJ)(t/T_(b)) containing only data dependent jitter and areconstructed signal {circumflex over (x)}_(PN(v))(t/T_(b)) containingonly vertical periodic noise are determined by the analysis module 20(steps S.3.3 and S.3.4).

Based on the reconstructed signals {circumflex over (x)}_(DDJ)(t/T_(b))and {circumflex over (x)}_(DDJ+PN(v))(t/T_(b)), the time interval errorTIE_(DDJ)(k) that is associated with data dependent jitter and the timeinterval error TIE_(DDJ+PJ(v))(k) that is associated with data dependentjitter and with vertical periodic jitter are determined (steps S.3.3.1and S.3.4.1).

Histograms

The analysis module 20 is configured to determine histograms of at leastone component of the time interval error based on the corresponding timeinterval error (step S.3.5).

Generally speaking, the analysis unit 20 is firstly configured todetermine the time interval error TIE_(Jx) associated with a jittercomponent Jx. The analysis module 20 can determine a histogramassociated with that jitter component Jx and may display it on thedisplay 22.

FIGS. 5A-5D show four examples of histograms determined by the analysismodule 20 that correspond to total jitter, data dependent jitter,periodic jitter and random jitter, respectively.

In the cases of total jitter and data dependent jitter, rising signaledges and falling signal edges are treated separately such thatinformation on duty cycle distortion is contained within the histogram.

Of course, a histogram corresponding only to certain components of theperiodic jitter and/or of the random jitter may be determined anddisplayed, in particular a histogram corresponding to at least one ofhorizontal periodic jitter, vertical periodic jitter, horizontal randomjitter and vertical random jitter.

Note that from FIG. 5D it can readily be seen that the time intervalerror associated with the random jitter is Gaussian-distributed.

Moreover, the deterministic jitter and the random jitter arestatistically independent from each other. Thus, the histogram of thetotal jitter may be determined by convolution of the histograms relatedto deterministic jitter and random jitter.

The measurement instrument 12 may be configured to selectively displayone or more of the determined histograms on the display 22.

In some embodiments, the user may choose which of the jitter componentsare selectively displayed.

Thus, the histogram corresponding to the time interval error associatedwith at least one of the vertical periodic jitter, the horizontalperiodic jitter, the vertical random jitter, the horizontal randomjitter, the data dependent jitter and the other bounded uncorrelatedjitter may be selectively displayed on the display 22.

Separation of Random Jitter and Horizontal Periodic Jitter

The analysis module 20 is configured to determine the time intervalerror TIE_(RJ) that is associated with the random jitter and the timeinterval error TIE_(PJ(h)) that is associated with the horizontalperiodic jitter (step S.3.6).

As shown in FIG. 3, the total time interval error TIE_(TJ)(k) and thetime interval error TIE_(DDJ+PJ(v)) that is associated with datadependent jitter and with vertical periodic jitter are determinedfirstly as already described above.

Then, TIE_(DDJ+PJ(v)) is subtracted from the total time interval errorTIE_(TJ)(k) such that the time interval error TIE_(RJ+PJ(h)) is obtainedthat only contains random jitter, horizontal periodic jitter and otherbounded uncorrelated jitter. In this regard, reference is made to FIG. 2illustrating an overview of the several jitter components.

Note that in the following, the other bounded uncorrelated jittercomponent is neglected. However, it may also be incorporated into theanalysis described below.

Analogously to the joint analysis method described above (step S.3.2),also the horizontal periodic jitter defined by the first term inequation (E.2), for example its time interval error, can be determinedby determining the corresponding amplitudes a_(i), frequencies ϑ_(i) andphases φ_(i). A flow chart of the corresponding method is depicted inFIG. 6.

For this purpose, the amplitudes a_(i), frequencies ϑ₁ and phases VP areroughly estimated at first (step S.6.1).

Then, at least these parameters are determined jointly (step S.6.2).

The time interval error TIE_(PJ(h)) that is associated with horizontalperiodic jitter is then reconstructed (step S.6.3). The result is givenby

$\begin{matrix}{{T\hat{I}{E_{{PJ}{(h)}}(k)}} = {\sum\limits_{i = 0}^{{\hat{N}}_{{PJ}{(h)}} - 1}\;{{\hat{a}}_{i} \cdot {{\sin\left( {{2{\pi \cdot {\hat{\vartheta}}_{i}}\text{/}{f_{b} \cdot k}} + {\hat{\varphi}}_{i}} \right)}.}}}} & \left( {E{.13}} \right)\end{matrix}$

From this, also the time interval error TIE_(RJ) being associated onlywith random jitter is calculated by subtracting TIE_(PJ(h)) fromTIE_(RJ+PJ(h)).

Determination of Random Jitter

Generally, the analysis module 20 is configured to determine astatistical moment that is associated with the temporal random jitterε_(RJ). Therein, the statistical moment is of second order or higher.

In some embodiments, the analysis module 20 is configured to determinethe variance o-RJ that is associated with the temporal random jitterε_(RJ) (step S.3.7). This step is explained in more detail below.

The approach is based on determining an autocorrelation functionr_(TIE,TIE)(m) of the time interval error that is defined by

${{r_{{TIE},{TIE}}(m)} = {\frac{1}{N_{ACF}(m)}{\sum\limits_{k = 0}^{{N_{ACF}{(m)}} - 1}\;{{{TIE}(k)} \cdot {{TIE}\left( {k + m} \right)}}}}},{m = 0},1,\ldots\;,{L_{ACF} - 1}$

wherein N_(ACF)(m) is the number of elements that are taken into accountfor the calculation of the autocorrelation function. As shown, thenumber of elements depends on displacement parameter m.

Further, L_(ACF) corresponds to the length of the autocorrelationfunction. The length may be adjustable by the user and/or may be equalto or bigger than the maximum of the maximal period of the periodicjitter and the length of the impulse response of the signal source 24and the transmission channel 26.

In general, the analysis module 20 may be configured to selectivelydetermine the respective autocorrelation function r_(TIE) _(Jx) _(,TIE)_(Jx) (m) of any jitter component Jx.

Generally, the measurement instrument 12 may be configured toselectively display the autocorrelation function r_(TIE) _(Jx) _(,TIE)_(Jx) (m) obtained on the display 22.

Accordingly, the approach for determining the variance σ_(ε) _(RJ) ²,namely the variance of the temporal random jitter ε_(RJ), is based ondetermining the autocorrelation function r_(TIE) _(RJ) _(,TIE) _(RJ) (m)of the random jitter.

A component x_(DDJ+RJ)(t/T_(b))≈x_(DDJ)(t/T_(b))+n_(RJ)(t/T_(b)) of theinput signal contains the data dependent jitter signal

$\begin{matrix}{{x_{DDJ}\left( {t\text{/}T_{b}} \right)} = {{\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h_{s}\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}}} + x_{- \infty}}} & \left( {E{.14}} \right)\end{matrix}$

and the perturbation

$\begin{matrix}{{n_{RJ}\left( {t\text{/}T_{b}} \right)} = {- {\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{{ɛ_{RJ}(k)}\text{/}{T_{b} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}}}}}} & \left( {E{.15}} \right)\end{matrix}$

that is caused by the random jitter ε_(RJ)(k)/T_(b). As shown above, theperiodic jitter is not taken into account in the following. However, itmight be taken into account if desired.

As can be seen from FIG. 7A, the time interval error TIE_(RJ) that isassociated with the random jitter ε_(RJ)(k)/T_(b) is given by

$\begin{matrix}{\frac{{TIE}_{RJ}\left( {t_{edge}\text{/}T_{b}} \right)}{T_{b}} \approx {{- \left\lbrack \frac{{dx}_{DDJ}\left( {t_{edge}\text{/}T_{b}} \right)}{d\left( {t\text{/}T_{b}} \right)} \right\rbrack^{- 1}} \cdot {{n_{RJ}\left( {t_{edge}\text{/}T_{b}} \right)}.}}} & \left( {E{.16}} \right)\end{matrix}$

In this approach the times t_(edge) of the signal edges of the datadependent jitter signal x_(DDJ)(t/T_(b)) are determined by the analysismodule 20, in particular based on the reconstructed data dependentjitter signal {circumflex over (x)}_(DDJ)(t/T_(b)).

Alternatively, as depicted in FIG. 7B the clock times t_(CLK) can beused that are known from the clock data recovery explained above (stepS.4.1). In this case, the time interval error TIE_(RJ) is given by

$\begin{matrix}{\frac{{TIE}_{RJ}\left( {t_{CLK}\text{/}T_{b}} \right)}{T_{b}} \approx {{- \left\lbrack \frac{{dx}_{DDJ}\left( {t_{CLK}\text{/}T_{b}} \right)}{d\left( {t\text{/}T_{b}} \right)} \right\rbrack^{- 1}} \cdot {n_{RJ}\left( {t_{CLK}\text{/}T_{b}} \right)}}} & \left( {E{.17}} \right)\end{matrix}$

In some embodiments, the clock times t_(CLK) can be used provided thatthe slopes of the respective jitter signals, namely the data dependentjitter signal x_(DDJ)(t/T_(b)) as well as the componentx_(DDJ+RJ)(t/T_(b)) of the input signal, are substantially equal asindicated in FIG. 7B.

The respective equations can be easily determined from the respectivegradient triangle in FIGS. 7A and 7B.

In the following, the relation of equation (E.16) is used to derive thevariance σ_(ε) _(RJ) ². However, it is to be understood that therelation of equation (E.17) could be used instead.

Using equation (E.16), the autocorrelation function of the random jitteris given by

$\begin{matrix}{{r_{{TIE}_{RJ},{TIE}_{RJ}}(m)} = {{E\left\{ {{{TIE}_{RJ}\left( {t_{edge}\text{/}T_{b}} \right)}\text{/}{T_{b} \cdot {{TIE}_{RJ}\left( {{t_{edge}\text{/}T_{b}} + m} \right)}}\text{/}T_{b}} \right\}} \approx {E{\left\{ {\left\lbrack \frac{{dx}_{DDJ}\left( {t_{CLK}\text{/}T_{b}} \right)}{d\left( \frac{t}{T_{b}} \right)} \right\rbrack^{- 1} \cdot \left\lbrack \frac{{dx}_{DDJ}\left( {{t_{CLK}\text{/}T_{b}} + m} \right)}{d\left( \frac{t}{T_{b}} \right)} \right\rbrack^{- 1} \cdot {n_{RJ}\left( {t_{CLK}\text{/}T_{b}} \right)} \cdot {n_{RJ}\left( {{t_{CLK}\text{/}T_{b}} + m} \right)}} \right\}.}}}} & \left( {E{.18}} \right)\end{matrix}$

Therein and in the following, E{y} indicates an expected value ofquantity y.

The method for determining the autocorrelation function r_(TIE) _(RJ)_(,TIE) _(RJ) is illustrated in FIG. 8.

The upper two rows in FIG. 8 represent a memory range of thetransmission channel 26 with a bit change at time k=0. Accordingly, thelower two rows represent a memory range of the transmission channel 26with a bit change at time k=m. Note that the example in FIG. 8 is for aPAM-2 coded input signal. However, the steps outlined in the followingcan readily be applied to a PAM-n coded input signal with appropriatecombinatorial changes.

The memory range of the transmission channel 26 is N_(pre)+N_(post)+1.Thus, there are 2^(N) ^(pre) ^(+N) ^(post) possible permutations {b(k)}of the bit sequence b(k) within the memory range.

The upper rows and the lower rows overlap in an overlap region startingat k=N_(start) and ending at k=N_(end). In the overlap region, thepermutations of the bit sequences b(k) in the memory ranges of the upperand the lower rows have to be identical.

Note that only the overlap region contributes to the autocorrelationfunction.

In order to calculate the number of possible permutations in the overlapregion, a case differentiation is made as follows:

The bit change at k=0 may be completely within the overlap region,completely outside of the overlap region or may overlap with the edge ofthe overlap region (i.e. one bit is inside of the overlap region and onebit is outside of the overlap region).

Similarly, the bit change at k=m may be completely within the overlapregion, completely outside of the overlap region or may overlap with theedge of the overlap region (i.e. one bit is inside of the overlap regionand one bit is outside of the overlap region).

Thus, there is a total of 3·3=9 cases that are taken into account.

Each permutation {b(k)} has a chance of P(u, v) for occurring andimplies a particular slope

${{{{dx}_{DDJ}\left( {t_{edge}\text{/}T_{b}} \right)}\text{/}{d\left( \frac{t}{T_{b}} \right)}}}_{({u,v})}$of the data dependent jitter signal x_(DDJ)(t_(edge)/T_(b)). Therein, uand v represent a particular realization of the bit sequence inside theoverlap region and a particular realization of the bit sequence outsideof the overlap region, respectively.

Now, the autocorrelation function of the perturbationn_(RJ)(t_(edge)/T_(b)+k) for two particular realizations of the memoryranges at times k=0 and k=m leading to the particular realization u inthe overlap region are determined. The conditional autocorrelationfunction of the perturbation n_(RJ)(t_(edge)/T_(b)) is determined to be

$\begin{matrix}{{{E\left\{ {{n_{RJ}\left( {t_{edge}\text{/}T_{b}} \right)} \cdot {n_{RJ}\left( {{t_{edge}\text{/}T_{b}} + m} \right)}} \right\}}}_{u} = {\sum\limits_{k_{0} = {- N_{pre}}}^{N_{post}}\;{\sum\limits_{k_{1} = {- N_{pre}}}^{N_{post}}\;{{\left\lbrack {{b\left( k_{0} \right)} - {b\left( {k_{0} - 1} \right)}} \right\rbrack \cdot \left\lbrack {{b\left( {k_{1} + m} \right)} - {b\left( {k_{1} - 1 + m} \right)}} \right\rbrack \cdot {h\left( {{{t_{edge}\text{/}T_{b}} - k_{0}},{b\left( k_{0} \right)}} \right)} \cdot {h\left( {{{t_{edge}\text{/}T_{b}} + m - k_{1}},{b\left( {k_{1} + m} \right)}} \right)} \cdot E}\left\{ {{ɛ_{RJ}\left( k_{0} \right)}\text{/}{T_{b} \cdot {ɛ_{RJ}\left( {k_{1} + m} \right)}}\text{/}T_{b}} \right\}}}}} & \left( {E{.19}} \right)\end{matrix}$

The temporal random jitter ε_(RJ)(k)/T_(b) is normally distributed, forexample stationary and normally distributed. Thus, the autocorrelationfunction for the temporal random jitter ε_(RJ)(k)/T_(b) can be isolatedsince the other terms relate to deterministic contributions. In fact,the autocorrelation function for the temporal random jitterε_(RJ)(k)/T_(b) is

$\begin{matrix}{{E\left\{ {{ɛ_{RJ}\left( k_{0} \right)}\text{/}{T_{b} \cdot {ɛ_{RJ}\left( {k_{1} + m} \right)}}\text{/}T_{b}} \right\}} = \left\{ {\begin{matrix}{\sigma_{\epsilon_{RJ}}^{2}\text{/}T_{b}^{2}} & {k_{0} = {k_{1} + m}} \\0 & {else}\end{matrix}.} \right.} & \left( {E{.20}} \right)\end{matrix}$

Hence, the autocorrelation function for the temporal random jitterε_(RJ)(k)/T_(b) has only one contribution different from zero, namelyfor k_(o)=k₁+m. Accordingly, equation (E.19) becomes

$\begin{matrix}{{{E\left\{ {{n_{RJ}\left( {t_{edge}\text{/}T_{b}} \right)} \cdot {n_{RJ}\left( {{t_{edge}\text{/}T_{b}} + m} \right)}} \right\}}}_{u} = {\frac{\sigma_{\epsilon_{RJ}}^{2}}{T_{b}^{2}} \cdot {\sum\limits_{k = N_{start}}^{N_{end}}\;{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack^{2} \cdot {h\left( {{{t_{edge}\text{/}T_{b}} - k},{b(k)}} \right)} \cdot {{h\left( {{{t_{edge}\text{/}T_{b}} + m - k},{b(k)}} \right)}.}}}}} & \left( {E{.21}} \right)\end{matrix}$

As already mentioned, only the overlap region has a contribution.Employing equation (E.21), the autocorrelation function of the randomjitter is determined to be

$\begin{matrix}{{{{r_{{TIE}_{RJ},{TIE}_{RJ}}(m)} \approx {\sum\limits_{u}{E\left\{ {{n_{RJ}\left( {t_{edge}\text{/}T_{b}} \right)} \cdot {n_{RJ}\left( {{t_{edge}\text{/}T_{b}} + m} \right)}} \right\}}}}❘_{u}{\cdot {\sum\limits_{v}{\sum\limits_{w}{{P\left( {\left( {u,v} \right)\bigcap\left( {u,w} \right)} \right)} \cdot \left\lbrack {\frac{{dx}_{DDJ}\left( {t_{edge}\text{/}T_{b}} \right)}{d\left( \frac{t}{T_{b}} \right)}❘_{({u,v})}} \right\rbrack^{- 1} \cdot \left\lbrack {\frac{{dx}_{DDJ}\left( {{t_{edge}\text{/}T_{b}} + m} \right)}{d\left( \frac{t}{T_{b}} \right)}❘_{({u,w})}} \right\rbrack^{- 1}}}}}},} & \left( {E{.22}} \right)\end{matrix}$

wherein P((u, v)∩(u, w)) is the joint probability density defined by

$\begin{matrix}{{P\left( {\left( {u,v} \right)\bigcap\left( {u,w} \right)} \right)} = {{{P\left( {u,v} \right)} \cdot {P\left( {\left( {u,w} \right)❘\left( {u,v} \right)} \right)}} = {{P\left( {u,v} \right)} \cdot {\frac{P\left( {u,w} \right)}{\sum\limits_{w}{{P\left( {u,w} \right)} \cdot {\sum\limits_{u}{P(u)}}}}.}}}} & \left( {E{.23}} \right)\end{matrix}$

As can clearly be seen from equations (E.21) and (E.22), theautocorrelation function r_(TIE) _(RJ) _(,TIE) _(RJ) (m) of the randomjitter is linearly dependent on the variance σ_(ϵ) _(RJ) ² of the randomjitter.

Thus, the variance σ_(∈) _(RJ) ² of the random jitter is determined asfollows.

On one hand, the impulse response h(t_(edge)/T_(b)−k,b(k)) is alreadyknown or can be determined, as it is the time derivative of thedetermined step response h_(s)(t/T_(b)−k,b(k)) evaluated at timet=t_(edge). Moreover, the bit sequence b(k) is also known via the signaldecoding procedure described above.

On the other hand, the time interval error TIE_(RJ)(k) is known from theseparation of the random jitter and the horizontal periodic jitterdescribed above (step S.3.6) and the autocorrelation function can bealso calculated from this directly.

Thus, the only unknown quantity in equations (E.21) and (E.22) is thevariance σ_(∈) _(RJ) ² of the random jitter, which can thus bedetermined from these equations.

As shown in FIG. 3 as well as FIG. 9, the autocorrelation function canbe determined for any jitter component.

In FIG. 9, the autocorrelation functions for the total jitter signal,the periodic jitter signal, the data dependent jitter signal as well asthe random jitter are shown.

Generally, the respective result may be displayed on the display 22.

Power Spectral Density

The power spectral density R_(TIE,TIE)(f/f_(b)) of the time intervalerror is calculated based on the autocorrelation function by a Fourierseries, which reads

$\begin{matrix}{{R_{{TIE},{TIE}}\left( {f\text{/}f_{b}} \right)} = {\sum\limits_{{mn} = {{- L_{ACF}} + 1}}^{{+ L_{ACF}} - 1}\;{{r_{{TIE},{TIE}}(m)} \cdot {e^{{{- j} \cdot 2}{\pi \cdot f}\text{/}{f_{b} \cdot m}}.}}}} & \left( {E{.24}} \right)\end{matrix}$

The analysis module 20 may be configured to selectively determine thepower spectral density R_(TIE) _(Jx) _(,TIE) _(Jx) (m) of any jittercomponent Jx.

Moreover, the measurement instrument 12 may be configured to selectivelydisplay the power spectral density R_(TIE) _(Jx) _(,TIE) _(Jx) (m) onthe display 22.

As shown in FIG. 3 as well as FIG. 10, the power spectral density can bedetermined for any jitter component.

In FIG. 10, the power spectral densities for the total jitter signal,the periodic jitter signal, the data dependent jitter signal as well asthe random jitter are shown.

Generally, the respective result may be displayed on the display 22.

Bit Error Rate

The analysis module 20 is configured to determine the bit error rateBER(t/T_(b)) that is caused by the time interval error TIE_(DJ+RJ) beingassociated with the deterministic jitter and the random jitter, i.e.with the total jitter (step S.3.8).

A bit error occurs if the time interval error TIE_(DJ) being associatedwith the deterministic jitter and the time interval error TIE_(RJ) beingassociated with the random jitter fulfill one of the following twoconditions:

$\begin{matrix}{{{\frac{t}{T_{b}} < {{TIE}_{DJ} + {TIE}_{RJ}}},{0 \leq \frac{t}{T_{b}} \leq \frac{1}{2}}}{{\frac{t}{T_{b}} > {1 + {TIE}_{DJ} + {TIE}_{RJ}}},{\frac{1}{2} < \frac{t}{T_{b}} < 1}}} & \left( {E{.25}} \right)\end{matrix}$

Thus, based on the histogram of the time interval error TIE_(DJ)associated with deterministic jitter and based on the variance σ_(RJ) ²of the time interval error TIE_(RJ), the bit error rate BER(t/T_(b)) isdetermined as follows for times t/T_(b)<½:

$\begin{matrix}{{{BER}\left( \frac{t}{T_{b}} \right)} = {{{P_{rise} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{rise}} - 1}\;{{P_{{DJx},{rise}}(i)} \cdot {\int_{\frac{t}{T_{b}} - {{TIE}_{{DJ},{rise}}{(i)}}}^{\infty}{\frac{1}{\sqrt{2\pi} \cdot \sigma_{RJ}} \cdot e^{\frac{- {RJ}^{2}}{2\sigma_{RJ}^{2}}} \cdot {dRJ}}}}}} + {P_{fall} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{fall}} - 1}\;{{P_{{DJx},{fall}}(i)} \cdot {\int_{\frac{t}{T_{b}} - {{TIE}_{{DJ},{fall}}{(i)}}}^{\infty}{\frac{1}{\sqrt{2\pi} \cdot \sigma_{RJ}} \cdot e^{\frac{- {RJ}^{2}}{2\sigma_{RJ}^{2}}} \cdot {dRJ}}}}}}} = {{\frac{P_{rise}}{2} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{rise}} - 1}{{P_{{DJx},{rise}}(i)} \cdot {{erfc}\left( \frac{\frac{t}{T_{b}} - {{TIE}_{{DJ},{rise}}(i)}}{\sqrt{2} \cdot \sigma_{RJ}} \right)}}}} + {\frac{P_{fall}}{2} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{fall}} - 1}{{P_{{DJx},{fall}}(i)} \cdot {{{erfc}\left( \frac{\frac{t}{T_{b}} - {{TIE}_{{DJ},{fall}}(i)}}{\sqrt{2} \cdot \sigma_{RJ}} \right)}.}}}}}}} & \left( {E{.26}} \right)\end{matrix}$

For times ½<t/T_(b)<1, the bit error rate BER (t/T_(b)) is determined tobe

$\begin{matrix}{{{BER}\left( \frac{t}{T_{b}} \right)} = {{{P_{rise} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{rise}} - 1}\;{{P_{{DJx},{rise}}(i)} \cdot {\int_{- \infty}^{\frac{t}{T_{b}} - {{TIE}_{{DJ},{rise}}{(i)}} - 1}{\frac{1}{\sqrt{2\pi} \cdot \sigma_{RJ}} \cdot e^{\frac{- {RJ}^{2}}{2\sigma_{RJ}^{2}}} \cdot {dRJ}}}}}} + {P_{fall} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{fall}} - 1}\;{{P_{{DJx},{fall}}(i)} \cdot {\int_{- \infty}^{\frac{t}{T_{b}} - {{TIE}_{{DJ},{fall}}{(i)}} - 1}{\frac{1}{\sqrt{2\pi} \cdot \sigma_{RJ}} \cdot e^{\frac{- {RJ}^{2}}{2\sigma_{RJ}^{2}}} \cdot {dRJ}}}}}}} = {{P_{rise} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{rise}} - 1}\;{{P_{{DJx},{rise}}(i)} \cdot \left\lbrack {1 - {\frac{1}{2}{{erfc}\left( \frac{\frac{t}{T_{b}} - {{TIE}_{{DJ},{rise}}(i)} - 1}{\sqrt{2} \cdot \sigma_{RJ}} \right)}}} \right\rbrack}}} + {P_{fall} \cdot {\sum\limits_{i = 0}^{N_{{DJ},{fall}} - 1}{{P_{{DJx},{fall}}(i)} \cdot {\left\lbrack {1 - {\frac{1}{2}{{erfc}\left( \frac{\frac{t}{T_{b}} - {{TIE}_{{DJ},{fall}}(i)} - 1}{\sqrt{2} \cdot \sigma_{RJ}} \right)}}} \right\rbrack.}}}}}}} & \left( {E{.27}} \right)\end{matrix}$

Therein, P_(rise) and P_(fall) are the probabilities of a rising signaledge and of a falling signal edge, respectively. N_(DJ,rise) andN_(DJ,fall) are the numbers of histogram containers of the deterministicjitter for rising signal edges and for falling signal edges,respectively. Correspondingly, TIE_(DJ,rise)(i) and TIE_(DJ,fall)(i) arethe histogram values of the deterministic jitter for rising signal edgesand for falling signal edges, respectively.

Thus, the bit error rate BER(t/T_(b)) is calculated based on thehistogram of the deterministic jitter and based on the variance of therandom jitter rather than determined directly by measuring the number ofbit errors occurring within a certain number of bits.

Generally spoken, the bit error rate BER(t/T_(b)) is determined based onthe respective time interval error used for deriving at thecorresponding histogram.

This way, the bit error rate can also be determined in regions that arenot accessible via direct measurements or that simply take a too longtime to measure, for example for bit error rates BER(t/T_(b))<10⁻⁶.

In some embodiments, bit error rates smaller than 10⁻⁸, smaller than10⁻¹⁰ or even smaller than 10⁻¹² can be determined employing the methoddescribed above.

In order to linearize the curves describing the bit error rate, amathematical scale transformation Q(t/T_(b)) may be applied to the biterror rate, which is, at least for the case of P_(rise)+P_(fall)=0.5,given by:Q(t/T _(b))=√{square root over (2)}·erf⁻¹(1−2·BER(t/T _(b)))   (E.28)

Instead of employing the histogram of the complete deterministic jitter,a histogram corresponding to at least one of the components of thedeterministic jitter may be employed. Put differently, one or more ofthe components of the deterministic jitter may be selectively suppressedand the corresponding change of the bit error rate may be determined.This is also shown in FIG. 3.

More precisely, one of or an arbitrary sum of the data dependent jitter,the other bounded uncorrelated jitter, the horizontal periodic jitterand the vertical periodic jitter may be included and the remainingcomponents of the deterministic jitter may be suppressed.

For instance, the bit error rate BER(t/T_(b)) is determined based on thehistogram related to data dependent jitter, the histogram related todata dependent jitter and periodic jitter or the histogram related todata dependent jitter and other bounded uncorrelated jitter.

Moreover, the horizontal and vertical components may be selectivelytaken into account. For example, the precision or rather accuracy may beimproved.

Analogously, only the variance of the vertical random jitter or of thehorizontal random jitter may be employed instead of the variance of thecomplete random jitter such that the other one of the two random jittercomponents is suppressed and the effect of this suppression may bedetermined.

The respective histograms may be combined in any manner. Hence, theperiodic jitter may be obtained by subtracting the data dependent jitterfrom the deterministic jitter.

Depending on which of the deterministic jitter components is included,the final result for the bit error rate BER(t/T_(b)) includes only thecontributions of these deterministic jitter components.

Thus, the bit error rate BER(t/T_(b)) corresponding to certain jittercomponents can selectively be determined.

The determined bit error rate BER(t/T_(b)) may be displayed on thedisplay 22 as shown in FIG. 11.

In FIG. 11, a measured bit error rate as well as a bit error rateestimated with methods known in the prior art are also shown.

In FIG. 12, the respective mathematical scale transformation is shownthat may also be displayed on the display 22.

In some embodiments, a bit error rate BER(t/T_(b)) containing onlycertain deterministic jitter components may be displayed on the display22, wherein a user may choose which of the deterministic jittercomponents are included. Moreover, the fraction of the complete biterror rate that is due to the individual jitter components may bedetermined and displayed on the display 22.

Note that if the individual histograms of two statistically independentcomponents TIE₀ and TIE₁ of the time interval error TIE are known, theresulting collective histogram containing both components can bedetermined by a convolution of the two individual histograms:

$\begin{matrix}{{f_{{TIE}_{0} + {TIE}_{1}}\left( {{TIE}_{0} + {TIE}_{1}} \right)} = {\sum\limits_{\xi = {- \infty}}^{+ \infty}\;{{f_{{TIE}_{0}}(\xi)} \cdot {{f_{{TIE}_{1}}\left( {{TIE}_{0} + {TIE}_{1} - \xi} \right)}.}}}} & \left( {E{.29}} \right)\end{matrix}$

As mentioned already, the deterministic jitter and the random jitter arestatistically independent from each other.

Thus, the histogram of the time interval error related to total jittermay be determined by convolution of the histograms of the time intervalerrors related to deterministic jitter and random jitter.

Joint random jitter and random noise analysis

The analysis module 20 is configured to separate the vertical randomnoise and the horizontal random jitter contained within the inputsignal.

More precisely, the analysis module is configured to perform a jointrandom jitter and random noise analysis of the input signal in order toseparate and/or determine the vertical random noise and the horizontalrandom jitter.

First, the determined data dependent jitter signal x_(DDJ)(t/T_(b)),which is determined in step S.4.4, and the determined vertical periodicnoise signal x_(PN(v)) (step S.3.2) are subtracted from the inputsignal, labelled in the following by x_(TJ)(t/T_(b)), thereby generatinga perturbation signal n₀(t/T_(b)), which is determined to be

$\begin{matrix}{{n_{0}\left( {t\text{/}T_{b}} \right)} = {{{x_{TJ}\left( {t\text{/}T_{b}} \right)} - {x_{DDJ}\left( {t\text{/}T_{b}} \right)} - {x_{{PN}{(v)}}\left( {t\text{/}T_{b}} \right)}} = {{- {\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{{ɛ(k)}\text{/}{T_{b} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}}}}} + {{x_{{RN}{(v)}}\left( {t\text{/}T_{b}} \right)}.}}}} & \left( {E{.30}} \right)\end{matrix}$

The perturbation signal n₀(t/T_(b)) approximately only containshorizontal random jitter, vertical random noises x_(RN(v))(t/T_(b)) andhorizontal periodic jitter, wherein the temporal jitter

$\begin{matrix}{{{ɛ(k)}\text{/}T_{b}} = {{{{ɛ_{PJ}(k)}\text{/}T_{b}} + {{ɛ_{RJ}(k)}\text{/}T_{b}}} = {{\sum\limits_{i = 0}^{N_{{PJ}{(h)}} - 1}\;{a_{i}\text{/}{T_{b} \cdot {\sin\left( {{2{\pi \cdot \vartheta_{i}}\text{/}{f_{b} \cdot k}} + \varphi_{i}} \right)}}}} + {{ɛ_{RJ}(k)}\text{/}T_{b}}}}} & \left( {E{.31}} \right)\end{matrix}$

approximately only contains horizontal random jitter and horizontalperiodic jitter.

As already mentioned, more than a single bit period may be taken intoaccount.

The next step performed by the analysis module 20 is to determine thehorizontal periodic jitter components.

For this purpose, a time variant equalizer filter ĥ_(e)(k,t/T_(b)) isapplied to the perturbation signal n₀(t/T_(b)). The time variantequalizer filter ĥ_(e)(k,t/T_(b)) is determined from a time variantequalizer filter {tilde over (h)}_(e)(k,t/T_(b)) that is defined by:{tilde over (h)} _(e)(k,t/T _(b))=[b(−k)−b(−k+1)]·h(k−t/T _(b) ,b(k)).  (E.32)

More precisely, the time variant equalizer filter is determined byminimizing the following cost functional K, in particular by applying aleast mean squares approach:

$\begin{matrix}{K = {\sum\limits_{f\text{/}f_{b}}{{\frac{1}{{\overset{\sim}{H}}_{e}\left( {f\text{/}f_{b}} \right)} - {\sum\limits_{k}{{{\hat{h}}_{e}(k)} \cdot e^{{- j}\; 2{\pi \cdot f}\text{/}{f_{b} \cdot k}}}}}}^{2}}} & \left( {E{.33}} \right)\end{matrix}$

Therein, {tilde over (H)}_(e)(f/f_(b)) is the Fourier transform of thetime variant equalizer filter {tilde over (h)}_(e)(k,t/T_(b)). Ofcourse, this analysis could also be performed in time domain instead ofthe frequency domain as in equation (E.33).

The resulting time variant equalizer filter is then applied to theperturbation signal n₀(t/T_(b)) such that a filtered perturbation signalis obtained, which is determined to be

$\begin{matrix}\begin{matrix}{{{\overset{\sim}{n}}_{0}\left( {k,{t\text{/}T_{b}}} \right)} = {{{ɛ_{PJ}(k)}\text{/}T_{b}} + {{{\overset{\sim}{ɛ}}_{RJ}(k)}\text{/}T_{b}}}} \\{= {\sum\limits_{k}{{{\hat{h}}_{e}\left( {k,{t\text{/}T_{b}}} \right)} \cdot {n_{0}\left( {{t\text{/}T_{b}} - k} \right)}}}} \\{= {{\sum\limits_{i = 0}^{N_{{PJ}{(h)}} - 1}\;{a_{i}\text{/}{T_{b} \cdot {\sin\left( {{2{\pi \cdot \vartheta_{i}}\text{/}{f_{b} \cdot k}} + \varphi_{i}} \right)}}}} +}} \\{{{{\overset{\sim}{ɛ}}_{{{RJ}{(h)}},{{RN}{(v)}}}(k)}\text{/}T_{b}},}\end{matrix} & \left( {E{.34}} \right)\end{matrix}$

Now, the frequencies ϑ_(i) and the phases φ_(i) are roughly estimated atfirst and then the amplitudes â_(i), the frequencies {circumflex over(ϑ)}_(i) and the phases {circumflex over (φ)}_(i) are determinedjointly. For this purpose, the following cost functional

$\begin{matrix}{K = {\sum\limits_{t\text{/}T_{b}}\left\lbrack {{n_{0}\left( {t\text{/}T_{b}} \right)} + {\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{\sum\limits_{i = 0}^{N_{{PJ}{(h)}} - 1}\;{\frac{{\hat{a}}_{i}}{T_{b}} \cdot {\sin\left( {{2{\pi \cdot {\hat{\vartheta}}_{i}}\text{/}{f_{b} \cdot k}} + {\hat{\varphi}}_{i}} \right)} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}}}}} \right\rbrack^{2}}} & \left( {E{.35}} \right)\end{matrix}$

is minimized analogously to the joint parameter analysis method outlinedabove that corresponds to step S.3.2. shown in FIG. 3.

If there is no duty cycle distortion or if the duty cycle distortionpresent in the input signal is much smaller than the horizontal periodicjitter, a time invariant equalizer filter {tilde over(h)}_(e)(k,t/T_(b))=h(k−t/T_(b)) may be used for determining the timeinvariant equalizer filter ĥ_(e)(k,t/T_(b)).

In this case, the filtered perturbation signal

$\begin{matrix}\begin{matrix}{{{\overset{\sim}{n}}_{0}\left( {k,{t\text{/}T_{b}}} \right)} = {{{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {ɛ_{PJ}(k)}}\text{/}T_{b}} + {{{\overset{\sim}{ɛ}}_{RJ}(k)}\text{/}T_{b}}}} \\{= {\sum\limits_{k}{{{\hat{h}}_{e}\left( {k,{t\text{/}T_{b}}} \right)} \cdot {n_{0}\left( {{t\text{/}T_{b}} - k} \right)}}}} \\{= {\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {\sum\limits_{i = 0}^{N_{{PJ}{(h)}} - 1}\;{a_{i}\text{/}{T_{b} \cdot}}}}} \\{{{\sin\left( {{2{\pi \cdot \vartheta_{i}}\text{/}{f_{b} \cdot k}} + \varphi_{i}} \right)} + {{{\overset{\sim}{ɛ}}_{{{RJ}{(h)}},{{RN}{(v)}}}(k)}\text{/}T_{b}}},}\end{matrix} & \left( {E{.36}} \right)\end{matrix}$

still comprises a modulation [b(k)−b(k−1)] that is due to the bitsequence, which is however known and is removed before roughlyestimating the frequencies ϑ_(i) and the phases φ_(i).

With the determined amplitudes â_(i), the determined frequencies{circumflex over (ϑ)}_(i) and the determined phases {circumflex over(φ)}_(i), the horizontal periodic jitter signal is now reconstructed tobe

$\begin{matrix}{{{\hat{x}}_{{PJ}{(h)}}\left( {t\text{/}T_{b}} \right)} = {- {\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{\sum\limits_{i = 0}^{N_{{PJ}{(h)}} - 1}\;{{\hat{a}}_{i}\text{/}{T_{b} \cdot {\sin\left( {{2{\pi \cdot {\hat{\vartheta}}_{i}}\text{/}{f_{b} \cdot k}} + {\hat{\varphi}}_{i}} \right)} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {{h\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}.}}}}}}} & \left( {E{.37}} \right)\end{matrix}$

Based on the reconstructed horizontal periodic jitter signal, a randomperturbation signal n₁(t/T_(b)) is determined by subtracting thereconstructed horizontal periodic jitter signal shown in equation (E.37)from the perturbation signal. The determined random perturbation signalreads

$\begin{matrix}{{n_{1}\left( {t\text{/}T_{b}} \right)} = {{{n\left( {t\text{/}T_{b}} \right)} - {{\hat{x}}_{{PJ}{(h)}}\left( {t\text{/}T_{b}} \right)}} \approx {{- {\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{{ɛ_{RJ}(k)}\text{/}{T_{b} \cdot \left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack \cdot {h\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}}}}} + {x_{{RN}{(v)}}\left( {t\text{/}T_{b}} \right)}}}} & \left( {E{.38}} \right)\end{matrix}$

and contains approximately only horizontal random jitter represented bythe first term in the second line of equation (E.38) and vertical randomnoise represented by the second term in the second line of equation(E.38).

Generally speaking, the analysis module 20 now applies a statisticalmethod to the signal of equation (E.38) at two different times in orderto determine two statistical moments that are associated with thehorizontal random jitter and with the vertical random noise,respectively.

More specifically, the analysis module 20 determines the variancesσ_(RJ(h)) ² and σ_(RN(v)) ² that are associated with the horizontalrandom jitter and with the vertical random noise, respectively, based onequation (E.38). Note that both the horizontal random jitter and thevertical random noise are normal-distributed. Further, they arestatistically independent from each other.

According to a first variant, the conditional expected value of n₁²(t/T_(b)) for a particular realization (u,v) of the memory range isused and is determined to be

$\begin{matrix}{{{E\left\{ {n_{1}^{2}\left( {t\text{/}T_{b}} \right)} \right\}}}_{({u,v})} \approx {{\frac{\sigma_{\epsilon_{RJ}}^{2}}{T_{b}^{2}} \cdot {\sum\limits_{i = 0}^{N - 1}\;{P_{({u_{i},v_{i}})} \cdot {\sum\limits_{k = {- N_{pre}}}^{N_{post}}\;{{\left\lbrack {{b(k)} - {b\left( {k - 1} \right)}} \right\rbrack^{2} \cdot h^{2}}{\quad\left( {{{t\text{/}T_{b}} - k},{b(k)}} \right)}_{({u_{i},v_{i}})}}}}}} + {\sigma_{{RN}{(v)}}^{2}.}}} & \left( {E{.39}} \right)\end{matrix}$

Therein, P_((u) _(i) _(,v) _(i) ₎ is the probability of the permutation(u_(i), v_(i)). N is the number of permutations that are taken intoaccount. Thus, the accuracy and/or the computational time can be adpatedby varying N. In particular, a user may choose the number N.

According to a second variant, all possible permutations (u_(i), v_(i))are taken into account in equation (E.39), such that an unconditionalexpected value of n₁ ²(t/T_(b)) is obtained that reads

$\begin{matrix}{{{E\left\{ {n_{1}^{2}\left( {t\text{/}T_{b}} \right)} \right\}} \approx {{\frac{\sigma_{\epsilon_{RJ}}^{2}}{T_{b}^{2}} \cdot {\sum\limits_{k = {- N_{pre}}}^{N_{post}}{P_{r} \cdot 2^{2} \cdot {h_{r}^{2}\left( {{t\text{/}T_{b}} - k} \right)}}}} + {P_{f} \cdot 2^{2} \cdot {h_{f}^{2}\left( {{t\text{/}T_{b}} - k} \right)}} + \sigma_{{RN}{(v)}}^{2}}},} & \left( {E{.40}} \right)\end{matrix}$

Therein, P_(r) and P_(f) are the probabilities for a rising signal flankand for a falling signal flank, respectively.

If there is no duty cycle distortion present or if the duty cycledistortion is very small, equation (E.39) simplifies to

$\begin{matrix}{{E\left\{ {n_{1}^{2}\left( {t\text{/}T_{b}} \right)} \right\}} \approx {{\frac{\sigma_{\epsilon_{RJ}}^{2}}{T_{b}^{2}} \cdot \left\lbrack {{2E\left\{ {b^{2}(k)} \right\}} - {2E\left\{ {{b(k)} \cdot {b\left( {k - 1} \right)}} \right\}}} \right\rbrack \cdot {\sum\limits_{k = {- N_{pre}}}^{N_{post}}{h^{2}\left( {{t\text{/}T_{b}} - k} \right)}}} + {\sigma_{{RN}{(v)}}^{2}.}}} & \left( {E{.41}} \right)\end{matrix}$

The analysis module 20 is configured to determine the variancesσ_(RJ(h)) ² and σ_(RN(v)) ² based on at least one of equations (E.39) to(E.41). More precisely, the respective equation is evaluated for atleast two different times t/T_(b). For example, the signal edge timet₀/T_(b)=0 and the time t₁/T_(b)=½ may be chosen.

As everything except for the two variances is known in equations (E.39)to (E.41), the variances σ_(RJ(h)) ² and σ_(RN(v)) ² are then determinedfrom the resulting at least two equations. It is to be noted that thevariances σ_(RJ(h)) ² and σ_(RN(v)) ² correspond to the respectivestandard deviations.

In order to enhance accuracy, the equations can be evaluated at morethan two times and fitted to match the measurement points in an optimalfashion, for example by applying a least mean squares approach.

Alternatively or additionally, only the variance σ_(RN(v)) ² may bedetermined from the equations above and the variance σ_(RJ(v)) ² may bedetermined from the following relation 2

$\begin{matrix}{{\sigma_{{RJ}{(v)}}^{2}\text{/}T_{b}^{2}} = {\sigma_{{RN}{(v)}}^{2} \cdot {\sum\limits_{i}{P_{i} \cdot \left\lbrack \frac{1}{{{dx}_{{DDJ}_{i}}\left( {t_{edge}\text{/}T_{b}} \right)}\text{/}\left( {{dt}\text{/}T_{b}} \right)} \right\rbrack^{2}}}}} & \left( {E{.42}} \right)\end{matrix}$

As the horizontal random jitter and the vertical random jitter arestatistically independent, the variance is then determined to beσ_(RJ(h)) ² /T _(b)=σ_(RJ) ² /T _(b) ²−σ_(RJ(v)) ² /T _(b) ²   (E.43)

Therein, P_(i) is the probability that a signal edge with slope dx_(DDJ)_(i) (t_(edge)/T_(b))/(dt/T_(b)) arises. The numerical complexity ofthis method can be reduced by only taking into account substantiallydifferent slopes to contribute to the sum of equation (E.42).

Separation of random jitter and other bounded uncorrelated jitter

The analysis module 20 is further configured to determine a probabilitydensity f_(x) ₀ (x₀) of a Gaussian random variable, for instance therandom jitter, and a probability density f_(x) ₁ (x₁) of a non-Gaussianbounded random variable, for instance the other bounded uncorrelatedjitter.

For instance, the separation of the random jitter and other boundeduncorrelated jitter may be done by modelling the random jitter x₀ with astandard deviation σ_(RJ) whereas the other bounded uncorrelated jitterx₁ is random having the probability density f_(x) ₁ (x₁).

The probability distribution may read as follows

${F_{x}(x)} = {{P_{0} \cdot \left\lbrack {1 - {\frac{1}{2}{{erfc}\left( \frac{x - \mu_{0}}{\sqrt{2} \cdot \sigma} \right)}}} \right\rbrack} + {P_{1} \cdot {\left\lbrack {1 - {\frac{1}{2}{{erfc}\left( \frac{x - \mu_{1}}{\sqrt{2} \cdot \sigma} \right)}}} \right\rbrack.}}}$

A mathematical scale transformation Q_(x)(x) as already described may beapplied so that

${Q_{x}(x)} = {{erfc}^{- 1}\left( {2 - {2 \cdot \left( {P_{0} + P_{1}} \right)} + {P_{0} \cdot {{erfc}\left( \frac{x - \mu_{0}}{\sqrt{2} \cdot \sigma} \right)}} + {P_{1} \cdot {{erfc}\left( \frac{x - \mu_{1}}{\sqrt{2} \cdot \sigma} \right)}}} \right)}$

is obtained, wherein the line obtained by the mathematical scaletransformation may correspond to

${{{{Q_{x}(x)}}_{left} \approx \frac{x - \mu_{0}}{\sqrt{2} \cdot \sigma}},{and}}\mspace{14mu}$${{Q_{x}(x)}\text{❘}_{right}} \approx \frac{x - \mu_{1}}{\sqrt{2} \cdot \sigma}$

for respective ends of the mathematical scale transformation.

The standard deviation σ and the parameters μ₀,μ₁ may be determined.

The standard deviation σ may also be determined differently, forinstance as already described above.

In the input signal, the random jitter and the other boundeduncorrelated jitter are superposed. Therefore, a collective probabilitydensity f_(x)(x) is given by a convolution of the individual probabilitydensities with x=x₀+x₁, i.e.

$\begin{matrix}{{f_{x}(x)} = {\int_{- \infty}^{+ \infty}{{{f_{x_{0}}(\xi)} \cdot {f_{x_{1}}\left( {x - \xi} \right)} \cdot d}\;\xi}}} & \left( {E{.43}} \right)\end{matrix}$

Transformed into frequency domain, the convolution of equation (E.43)becomes a mere product. The Fourier transform F_(x) ₀ (f/f_(a)), i.e.the spectrum of the random jitter probability density f_(x) ₀ (x₀), isnormal distributed and reads:

$\begin{matrix}{{F_{x_{0}}\left( {f\text{/}f_{a}} \right)} = {e^{{- 2}{\pi^{2} \cdot \overset{¨}{\sigma^{2}} \cdot {(\frac{f}{f_{a}})}^{2}}}.}} & \left( {E{.44}} \right)\end{matrix}$

This property is employed in the separation of the random jittercomponent and the other bounded uncorrelated jitter component.

For example, the spectrum is determined based on measurements of theinput signal and by matching the function of equation (E.44) to themeasurement data.

In FIG. 13, an overview is shown wherein the probability densities ofthe random jitter, the other bounded uncorrelated jitter as well as asuperposition of both are illustrated.

Alternatively or additionally, the variance σ_(RJ) ² of the randomjitter may already be known from one of the steps described above.

The probability density of the random jitter component is thendetermined to be

$\begin{matrix}{{{\hat{f}}_{x_{0}}\left( x_{0} \right)} = {\frac{1}{\sqrt{2\pi} \cdot \hat{\sigma}} \cdot e^{- \frac{x_{0}^{2}}{2{\hat{\sigma}}^{2}}}}} & \left( {E{.45}} \right)\end{matrix}$

Based on the result of equation (E.45), the probability density f_(x) ₁(x₁) of the other bounded uncorrelated jitter component is thendetermined by a deconvolution of equation (E.43).

This is achieved by minimizing the following cost functional K, inparticular via a least mean squares approach:

$K = {\sum\limits_{x = x_{\min}}^{x_{\max}}\;{\left\lbrack {{f_{x}(x)} - {\sum\limits_{\xi = x_{1,\min}}^{x_{1,\max}}\;{{{\hat{f}}_{x_{1}}(\xi)} \cdot {{\hat{f}}_{x_{0}}\left( {x - \xi} \right)}}}} \right\rbrack^{2}.}}$

Thus, the histogram of the other bounded uncorrelated jitter componentcan be determined.

Accordingly, histograms of all jitter components may be determined asalready mentioned and shown in FIG. 3.

Certain embodiments disclosed herein utilize circuitry (e.g., one ormore circuits) in order to implement protocols, methodologies ortechnologies disclosed herein, operably couple two or more components,generate information, process information, analyze information, generatesignals, encode/decode signals, convert signals, transmit and/or receivesignals, control other devices, etc. Circuitry of any type can be used.

In an embodiment, circuitry includes, among other things, one or morecomputing devices such as a processor (e.g., a microprocessor), acentral processing unit (CPU), a digital signal processor (DSP), anapplication-specific integrated circuit (ASIC), a field-programmablegate array (FPGA), a system on a chip (SoC), or the like, or anycombinations thereof, and can include discrete digital or analog circuitelements or electronics, or combinations thereof. In an embodiment,circuitry includes hardware circuit implementations (e.g.,implementations in analog circuitry, implementations in digitalcircuitry, and the like, and combinations thereof).

In an embodiment, circuitry includes combinations of circuits andcomputer program products having software or firmware instructionsstored on one or more computer readable memories that work together tocause a device to perform one or more protocols, methodologies ortechnologies described herein. In an embodiment, circuitry includescircuits, such as, for example, microprocessors or portions ofmicroprocessor, that require software, firmware, and the like foroperation. In an embodiment, circuitry includes an implementationcomprising one or more processors or portions thereof and accompanyingsoftware, firmware, hardware, and the like.

The present application may reference quantities and numbers. Unlessspecifically stated, such quantities and numbers are not to beconsidered restrictive, but exemplary of the possible quantities ornumbers associated with the present application. Also in this regard,the present application may use the term “plurality” to reference aquantity or number. In this regard, the term “plurality” is meant to beany number that is more than one, for example, two, three, four, five,etc. The terms “about,” “approximately,” “near,” etc., mean plus orminus 5% of the stated value. For the purposes of the presentdisclosure, the phrase “at least one of A and B” is equivalent to “Aand/or B” or vice versa, namely “A” alone, “B” alone or “A and B.”.Similarly, the phrase “at least one of A, B, and C,” for example, means(A), (B), (C), (A and B), (A and C), (B and C), or (A, B, and C),including all further possible permutations when greater than threeelements are listed.

The principles, representative embodiments, and modes of operation ofthe present disclosure have been described in the foregoing description.However, aspects of the present disclosure which are intended to beprotected are not to be construed as limited to the particularembodiments disclosed. Further, the embodiments described herein are tobe regarded as illustrative rather than restrictive. It will beappreciated that variations and changes may be made by others, andequivalents employed, without departing from the spirit of the presentdisclosure. Accordingly, it is expressly intended that all suchvariations, changes, and equivalents fall within the spirit and scope ofthe present disclosure, as claimed.

The embodiments of the invention in which an exclusive property orprivilege is claimed are defined as follows:
 1. A jitter decompositionmethod for decomposing several jitter and noise components contained inan input signal, wherein the input signal is generated by a signalsource, comprising: receiving said input signal; at least one ofdetermining and receiving a reconstructed data dependent jitter signal;at least one of determining and receiving an impulse response, theimpulse response being associated with at least said signal source; anddetermining at least a first statistical parameter being associated witha first jitter component or a first noise component in said input signaland a second statistical parameter being associated with a second jittercomponent or a second noise component in said input signal, the secondjitter component or the second noise component being different from thefirst jitter component or the first noise component, respectively,wherein the first statistical parameter and the second statisticalparameter are determined by applying a statistical method at least attwo different times, and wherein the first statistical parameter and thesecond statistical parameter are each determined based on at least oneof the reconstructed data dependent jitter signal and the impulseresponse.
 2. The jitter decomposition method of claim 1, wherein thefirst statistical parameter is associated with a vertical jittercomponent and the second statistical parameter is associated with ahorizontal jitter component, wherein the vertical jitter component iscaused by an amplitude perturbation, and wherein the horizontal jittercomponent is caused by a temporal perturbation.
 3. The jitterdecomposition method of claim 1, wherein the first statistical parameteris associated with a noise component and the second statisticalparameter is associated with a jitter component.
 4. The jitterdecomposition method of claim 1, wherein at least one jitter componentin said input signal is neglected or set to zero, or wherein at leastone noise component in said input signal is neglected or set to zero. 5.The jitter decomposition method of claim 1, wherein a vertical periodicnoise signal in said input signal is neglected.
 6. The jitterdecomposition method of claim 1, wherein the first statistical parameterand the second statistical parameter comprise a statistical moment ofsecond order or higher.
 7. The jitter decomposition method of claim 1,wherein said input signal is decoded, thereby generating a decoded inputsignal.
 8. The jitter decomposition method of claim 1, wherein thesecond statistical parameter is determined based on at least the firststatistical parameter.
 9. The jitter decomposition method of claim 1,wherein a third statistical parameter associated with a third jittercomponent or a third noise component in said input signal is determined,the third jitter component or the third noise component being differentfrom the first jitter component or the first noise component as well asthe second jitter component or the second noise component.
 10. Thejitter decomposition method of claim 1, wherein a reconstructed verticalperiodic noise signal is at least one of determined and received,wherein the vertical periodic noise signal is associated with periodicnoise that is caused by an amplitude perturbation.
 11. The jitterdecomposition method of claim 1, wherein a horizontal periodic jittersignal is determined based on at least one of the input signal, thereconstructed data dependent jitter signal and a reconstructed verticalperiodic noise signal, wherein the horizontal periodic jitter signal isassociated with periodic jitter that is caused by a temporalperturbation.
 12. The jitter decomposition method of claim 1, wherein arandom perturbation signal is determined based on at least one of theinput signal, the reconstructed data dependent jitter signal, areconstructed vertical periodic noise signal and a determined horizontalperiodic jitter signal.
 13. The jitter decomposition method of claim 12,wherein the second statistical parameter corresponds to a statisticalparameter assigned to at least one of the horizontal random jitter andthe vertical random jitter, wherein the second statistical parameter isdetermined based on the random perturbation signal, wherein thestatistical parameter assigned to the horizontal random jitter isassociated with random jitter that is caused by a temporal perturbation,and wherein the statistical parameter assigned to the vertical randomjitter is associated with random jitter that is caused by an amplitudeperturbation.
 14. The jitter decomposition method of claim 12, whereinthe first statistical parameter corresponds to a statistical parameterassigned to the vertical random noise, wherein the first statisticalparameter is determined based on the random perturbation signal, thefirst statistical parameter being associated with a vertical randomnoise component of said input signal, wherein the vertical periodicnoise component is associated with periodic noise that is caused by atemporal perturbation.
 15. The jitter decomposition method of claim 14,wherein the second statistical parameter corresponds to a statisticalparameter assigned to at least one of the horizontal random jitter andthe vertical random jitter, wherein the second statistical parameter isdetermined based on at least one of the random perturbation signal andthe determined first statistical parameter, wherein the statisticalparameter assigned to the horizontal random jitter is associated withrandom jitter that is caused by a temporal perturbation, and wherein thestatistical parameter assigned to the vertical random jitter isassociated with random jitter that is caused by an amplitudeperturbation.
 16. The jitter decomposition method of claim 11, whereinat least one of a time variant equalizer filter and a time invariantequalizer filter is applied in order to determine said horizontalperiodic jitter signal.
 17. The jitter decomposition method of claim 11,wherein horizontal periodic jitter signal parameters are determined inorder to determine said horizontal periodic jitter signal, thehorizontal periodic jitter signal parameters being associated withperiodic functions.
 18. The jitter decomposition method of claim 17,wherein the horizontal periodic jitter signal parameters are determinedjointly.
 19. The jitter decomposition method of claim 17, wherein thehorizontal periodic jitter signal parameters are determined by at leastone of minimizing and maximizing a cost functional.
 20. The jitterdecomposition method of claim 10, wherein the reconstructed datadependent jitter signal, the reconstructed vertical periodic noisesignal and the determined horizontal periodic jitter signal aresubtracted from the input signal in order to determine a randomperturbation signal.
 21. The jitter decomposition method of claim 20,wherein a statistical analysis of the random perturbation signal isperformed in order to determine at least one of the first statisticalparameter and the second statistical parameter.
 22. The jitterdecomposition method of claim 20, wherein an expected value of therandom perturbation signal squared is determined in order to determineat least one of the first statistical parameter and the secondstatistical parameter.
 23. The jitter decomposition method of claim 1,wherein said input signal is PAM-n coded, wherein n is an integer biggerthan
 1. 24. A measurement instrument, comprising: at least one inputchannel; and an analysis circuit being connected to the at least oneinput channel, wherein the measurement instrument being configured toreceive an input signal via said input channel and to forward the inputsignal to the analysis circuit, the analysis circuit being configured toat least one of determine and receive a reconstructed data dependentjitter signal; the analysis circuit being configured to at least one ofdetermine and receive an impulse response, the impulse response beingassociated with at least said signal source; and the analysis circuitbeing configured to determine at least a first statistical parameterbeing associated with a first jitter component or a first noisecomponent in said input signal and a second statistical parameter beingassociated with a second jitter component or a second noise component insaid input signal, wherein the first statistical parameter and thesecond statistical parameter are determined by applying a statisticalmethod at two different times, and wherein the first statisticalparameter and the second statistical parameter are each determined basedon at least one of the reconstructed data dependent jitter signal andthe impulse response.
 25. The measurement instrument of claim 24,wherein the first statistical parameter is associated with a verticaljitter component and the second statistical parameter is associated witha horizontal jitter component, wherein the vertical jitter component iscaused by an amplitude perturbation, and wherein the horizontal jittercomponent is caused by a temporal perturbation.
 26. The measurementinstrument of claim 24, wherein the first statistical parameter isassociated with a noise component and the second statistical parameteris associated with a jitter component.
 27. The measurement instrument ofclaim 24, wherein the analysis circuit is configured to neglect or setto zero at least one of a jitter component in said input signal, noisecomponent in said input signal and a vertical periodic noise signal insaid input signal.
 28. The measurement instrument of claim 24, whereinthe first statistical parameter and the second statistical parametercomprise a statistical moment of second order or higher.
 29. Themeasurement instrument of claim 24, wherein the analysis circuit isconfigured to decode said input signal, thereby generating a decodedinput signal.